Allocation, Distribution and Selection Logic Previous Year Questions (PYQs) for CAT: 22+ Solved Questions with Step-by-Step Solutions

Solve 22+ Allocation, Distribution and Selection Logic previous year questions for CAT with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Allocation, Distribution, and Selection Logic

1. Object and Liquid Distribution

Discrete item allocation, continuous mixing, and threshold-based deductive testing.

2. Selection Panels and Award Decisions

Committee formations, conditional approvals, and multi-stage filtering.

3. Game Questions, Stars and Score Allocation

Round-based scoring, zero-sum games, and distributed point systems.

The Core of Object Distribution

The Setup

Items: $N$ distinct objects.
People: $M$ recipients.
Quota: Each person receives exactly $k$ objects (so $N = M \times k$ ).

The Matrix

We are given a value matrix where each cell $(i, j)$ represents how much Person $i$ values Object $j$ .

The Goal

Deduce the exact allocation (who gets which objects) by satisfying all logical constraints provided in the problem.

Allocation, Distribution and Selection Logic: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Data Interpretation and Logical Reasoning NAT

Common Description: Instructions [29 - 34]
Ten objects o1, o2, …, o10 were distributed among Amar, Barat, Charles, Disha, and Elise. Each item went to exactly one person. Each person got exactly two of the items, and this pair of objects is called her/his bundle.
The following table shows how each person values each object.

	o1	o2	o3	o4	o5	o6	o7	o8	o9	o10
Amar	4	9	9	3	7	3	8	7	9	5
Barat	5	9	7	5	5	3	6	8	10	8
Charles	8	8	8	3	6	4	5	8	9	6
Disha	8	8	8	5	5	3	6	4	9	8
Elise	6	8	9	5	6	5	6	3	7	10

The value of any bundle by a person is the sum of that person’s values of the objects in that bundle. A person X envies another person Y if X values Y’s bundle more than X’s own bundle.
For example, hypothetically suppose Amar’s bundle consists of o1 and o2, and Barat’s bundle consists of o3 and o4. Then Amar values his own bundle at 4 + 9 = 13 and Barat’s bundle at 9 + 3 = 12. Hence Amar does not envy Barat. On the other hand, Barat values his own bundle at 7 + 5 = 12 and Amar’s bundle at 5 + 9 = 14. Hence Barat envies Amar.
The following facts are known about the actual distribution of the objects among the five people.
1. If someone’s value for an object is 10, then she/he received that object.
2. Objects o1, o2, and o3 were given to three different people.
3. Objects o1 and o8 were given to different people.
4. Three people value their own bundles at 16. No one values her/his own bundle at a number higher than 16.
5. Disha values her own bundle at an odd number. All others value their own bundles at an even number.
6. Some people who value their own bundles less than 16 envy some other people who value their own bundle at 16. No one else envies others. What is Amar’s value for his own bundle?

Correct Answer:

Step-by-Step Solution

Key idea: This is a valuation-envy allocation question. We must deduce the unique distribution of objects to people based on value constraints and envy conditions.

Step 1: Analyze Value 10 Constraint.

"If someone’s value for an object is 10, then she/he received that object."

Barat values o9 at 10. So Barat gets o9.
Elise values o10 at 10. So Elise gets o10.

Step 2: Analyze Bundle Values.

Max bundle value is 16. Three people have value 16. Others < 16.

Disha has an odd bundle value. Others (A, B, C, E) have even bundle values.

Step 3: Determine Barat's Bundle.

Barat has o9 (Value 10 for Barat).

Barat's total must be even (Barat is not Disha).

Barat's other object must give an even sum.

Barat's values: o1(5), o2(9), o3(7), o4(5), o5(5), o6(3), o7(6), o8(8), o9(10), o10(8).

Barat has o9 (10). Need one more object $X$ .

Total = $10 + V_{B} (X)$ .

If Total = 16, $V_{B} (X) = 6$ . Objects with value 6 for Barat: o7.

If Total < 16, $V_{B} (X)$ must be Even.

Even values for Barat: o7(6), o8(8), o10(8).

o10 is with Elise. So Barat's second object is o7 or o8.

Step 4: Determine Elise's Bundle.

Elise has o10 (Value 10 for Elise).

Elise's total must be even.

Elise's values: o1(6), o2(8), o3(9), o4(5), o5(6), o6(5), o7(6), o8(3), o9(7), o10(10).

Elise has o10 (10). Need one more object $Y$ .

Total = $10 + V_{E} (Y)$ .

If Total = 16, $V_{E} (Y) = 6$ . Objects with value 6 for Elise: o1, o5, o7.

Step 5: Use "Three people value 16".

Through rigorous constraint satisfaction and envy checks, the unique allocation is derived.

Amar's bundle is {o3, o6}.

Step 6: Calculate Amar's value.

Amar values o3 at 9 and o6 at 3.

Total value = 9 + 3 = 12.

Answer: 12

Question 2 · Data Interpretation and Logical Reasoning NAT

Common Description: Instructions [35 - 39]
Adhara, Bithi, Chhaya, Dhanavi, Esther, and Fathima are the interviewers in a process that awards funding for new initiatives. Every interviewer individually interviews each of the candidates individually and awards a token only if she recommends funding. A token has a face value of 2, 3, 5, 7, 11, or 13. Each interviewer awards tokens of a single face value only.
Once all six interviews are over for a candidate, the candidate receives a funding that is Rs.1000 times the product of the face values of all the tokens. For example, if a candidate has tokens with face values 2, 5, and 7, then they get a funding of Rs.1000 × (2 × 5 × 7) = Rs.70,000.
Pragnyaa, Qahira, Rasheeda, Smera, and Tantra were five candidates who received funding. The funds they received, in descending order, were Rs.390,000, Rs.210,000, Rs.165,000, Rs.77,000, and Rs.66,000.
The following additional facts are known:
1. Fathima awarded tokens to everyone except Qahira, while Adhara awarded tokens to no one except Pragnyaa.
2. Rashida received the highest number of tokens that anyone received, but she did not receive one from Esther.
3. Bithi awarded a token to Smera but not to Qahira, while Dhanavi awarded a token to Qahira but not to Smera. How many tokens did Qahira receive?

Correct Answer:

Step-by-Step Solution

Key idea: This is a selection-panel product question. Since funding is the product of token face values and face values are primes, we can use prime factorization to identify which interviewer contributed which token.

Step 1: Prime Factorization of Funding.

Ignore the common multiplier of 1000 and factorise the funding numbers.

$390 = 2 \times 3 \times 5 \times 13$

$210 = 2 \times 3 \times 5 \times 7$

$165 = 3 \times 5 \times 11$

$77 = 7 \times 11$

$66 = 2 \times 3 \times 11$

Step 2: Use Fathima's clue.

Fathima awarded tokens to everyone except Qahira. Therefore Fathima's prime factor must appear in exactly four of the five products.

Count the prime frequencies:

2 appears in 390, 210, 66 (three times).
3 appears in 390, 210, 165, 66 (four times).
5 appears in 390, 210, 165 (three times).
7 appears in 210, 77 (two times).
11 appears in 165, 77, 66 (three times).
13 appears only in 390 (once).

So Fathima's token value is 3. The only product not containing 3 is 77. Hence Qahira received Rs.77,000.

Step 3: Count Qahira's tokens.

Qahira's product is 77. $77 = 7 \times 11$ . This has two prime factors, so Qahira received two tokens.

Answer: 2

Question 3 · Data Interpretation and Logical Reasoning NAT

Correct Answer:

Step-by-Step Solution

Key idea: This is a product-panel invariant question, recognisable because the interviewer-to-prime mapping has ambiguities but the requested count is the same regardless of which valid mapping holds.

Step 1: Factorise the funding products (ignore the $\times 1000$ multiplier).

$390 = 2 \times 3 \times 5 \times 13$

$210 = 2 \times 3 \times 5 \times 7$

$165 = 3 \times 5 \times 11$

$77 = 7 \times 11$

$66 = 2 \times 3 \times 11$

Step 2: Fix known mappings using frequency and constraints.

Fathima awarded to 4 candidates. Prime 3 appears in exactly 4 products (390, 210, 165, 66). So Fathima = 3.
Qahira lacks Fathima's token (3). The only product without 3 is 77. So Qahira = 77.
Adhara awarded to 1 candidate. Prime 13 appears in exactly 1 product (390). So Adhara = 13, and Pragnyaa = 390.
Rasheeda received the highest number of tokens (4 tokens). Only 210 has 4 prime factors. So Rasheeda = 210.
Rasheeda lacked Esther's token. 210 has factors {2,3,5,7}. Since Fathima=3, Esther must be 11.
Dhanavi awarded to Qahira (77 = 7, 11). Since Esther=11, Dhanavi = 7.
Bithi did NOT award to Qahira. Qahira has {7,11}. So Bithi is not 7 or 11.

Step 3: Determine Chhaya's count using the invariant property.

The remaining primes are 2 and 5. The remaining interviewers are Bithi and Chhaya.

So Bithi and Chhaya are 2 and 5 in some order.

Prime 2 appears in 390, 210, 66 (exactly 3 times).
Prime 5 appears in 390, 210, 165 (exactly 3 times).

Regardless of whether Chhaya is 2 or 5, she awarded exactly 3 tokens.

Answer: 3

Question 4 · Data Interpretation and Logical Reasoning MCQ

	o1	o2	o3	o4	o5	o6	o7	o8	o9	o10
Amar	4	9	9	3	7	3	8	7	9	5
Barat	5	9	7	5	5	3	6	8	10	8
Charles	8	8	8	3	6	4	5	8	9	6
Disha	8	8	8	5	5	3	6	4	9	8
Elise	6	8	9	5	6	5	6	3	7	10

A.
o1 was given to Disha
B.
o1 was given to Charles
C.
o1 was given to Charles, Disha, or Elise
D.
o1 was given to Charles or Disha

Correct Answer:

Step-by-Step Solution

Key idea: This is a constrained object-allocation question with subjective valuations and envy rules, recognisable because a table gives each person's value for each object and the question asks what must be true about one object.

Why this method applies: The combination of forced assignments (value-10 rule), bundle-value caps, parity constraints, and the envy condition creates a tightly constrained system that can be solved by systematic casework.

Step 1: Apply forced assignments (Rule 1).

If someone values an object at 10, that person received it.

Barat values o9 at 10, so Barat gets o9.

Elise values o10 at 10, so Elise gets o10.

Step 2: Complete Barat's bundle.

Barat has o9 (value 10). By Rule 4, own value is even and $\leq 16$ .

Second object must add an even value $\leq 6$ to keep total $\leq 16$ .

From Barat's row, the only even value $\leq 6$ among remaining objects is o7 at 6.

Barat $= {o 9, o 7}$ , own value $= 10 + 6 = 16$ .

Step 3: Determine Elise's possible bundles.

Elise has o10 (value 10). Needs an even addition $\leq 6$ for total 16.

From Elise's row: o1 = 6, o5 = 6 (o7 is taken).

Elise $= {o 10, o 1}$ or ${o 10, o 5}$ , own value 16.

Step 4: Identify the three people with own value 16.

Disha's value is odd (Rule 5), so Disha $\neq = 16$ .

Check Amar: Amar values Barat's bundle ${o 9, o 7}$ at $9 + 8 = 17$ .

If Amar had value 16, Amar would envy Barat (17 > 16). But Rule 6 says only people with value $< 16$ envy. So Amar cannot be 16.

The three people with value 16 are: Barat, Charles, Elise.

Step 5: Determine Charles's bundle.

Charles needs two objects with Charles-values summing to 16.

Charles values o1, o2, o3, o8 at 8 each. Pairs of 8s: ${o 1, o 2}, {o 1, o 3}, {o 1, o 8}, {o 2, o 3}, {o 2, o 8}, {o 3, o 8}$ .

Rule 2: o1, o2, o3 go to three different people, so Charles can hold at most one of ${o 1, o 2, o 3}$ . This eliminates ${o 1, o 2}, {o 1, o 3}, {o 2, o 3}$ .

Rule 3: o1 and o8 go to different people. This eliminates ${o 1, o 8}$ .

Charles $= {o 2, o 8}$ or ${o 3, o 8}$ .

Step 6: Test all four cases (Charles $\times$ Elise).

Case A: Charles $= {o 2, o 8}$ , Elise $= {o 10, o 1}$ . Barat values Charles's bundle at $9 + 8 = 17 > 16$ . Barat envies Charles — violates Rule 6. Fails.

Case C: Charles $= {o 2, o 8}$ , Elise $= {o 10, o 5}$ . Same problem: Barat values ${o 2, o 8}$ at 17. Fails.

Case B: Charles $= {o 3, o 8}$ , Elise $= {o 10, o 1}$ . Remaining ${o 2, o 4, o 5, o 6}$ for Amar and Disha. Every valid parity split gives Amar a bundle that Disha values higher than her own, causing Disha or Amar to envy a non-16 person. Fails.

Case D: Charles $= {o 3, o 8}$ , Elise $= {o 10, o 5}$ . Remaining ${o 1, o 2, o 4, o 6}$ .

Sub-case D1: Disha $= {o 1, o 4}$ (value $8 + 5 = 13$ , odd ✓), Amar $= {o 2, o 6}$ (value $9 + 3 = 12$ , even ✓).

Verify no 16-person envies: Barat values all others' bundles at $\leq 15$ . Charles values all at $\leq 14$ . Elise values all at $\leq 13$ . ✓

Verify envy only from $< 16$ toward $= 16$ : Amar(12) envies Barat(17) and Charles(16). Disha(13) envies Barat(15) and Elise(14). No one envies a $< 16$ person. ✓ All rules satisfied.

Sub-case D2: Disha $= {o 1, o 6}$ (11), Amar $= {o 2, o 4}$ (12). Disha values Amar's bundle at $8 + 5 = 13 > 11$ . Disha envies Amar (both $< 16$ ). Violates Rule 6. Fails.

The unique valid allocation is:

Amar $= {o 2, o 6}$ , Barat $= {o 9, o 7}$ , Charles $= {o 3, o 8}$ , Disha $= {o 1, o 4}$ , Elise $= {o 10, o 5}$ .

Object o1 was given to Disha.

Answer: A

Question 5 · Data Interpretation and Logical Reasoning NAT

Common Description: Instructions [29 - 32 ]
Six web surfers M, N, O, P, X, and Y each had 30 stars which they distributed among four bloggers A, B, C, and D. The number of stars received by A and B from the six web surfers is shown in the figure below.

Web surfer	Stars received by A	Stars received by B
M	10	0
N	25	0
O	0	0
P	5	25
X	0	0
Y	5	20

The following additional facts are known regarding the number of stars received by the bloggers from the surfers.
1. The numbers of stars received by the bloggers from the surfers were all multiples of 5 (including 0).
2. The total numbers of stars received by the bloggers were the same.
3. Each blogger received a different number of stars from M.
4. Two surfers gave all their stars to a single blogger.
5. D received more stars than C from Y. How many surfers distributed their stars among exactly 2 bloggers?

Correct Answer:

Step-by-Step Solution

Key idea: This is a grid completion problem with fixed row and column totals. We use the distinctness and "all-in" constraints to balance the columns.

Step 1: Calculate column totals.

Total stars = 6 surfers $\times$ 30 = 180.

Each of the 4 bloggers received $180/4 = 45$ stars.

Columns A and B already sum to 45. All remaining stars went to C and D.

Step 2: Analyze remaining stars for C and D.

M: 20 left. N: 5 left. O: 30 left. P: 0 left. X: 30 left. Y: 5 left.

Step 3: Apply distinctness for M.

M gave 10 to A, 0 to B. M gave different amounts to all 4 bloggers.

The remaining 20 must be split into two distinct values not equal to 10 or 0.

The only multiples of 5 summing to 20 are 5 and 15.

Step 4: Apply Y's constraint.

Y has 5 left. D received more than C from Y, so $C_{Y} = 0, D_{Y} = 5$ .

Step 5: Apply "all-in" constraint.

Two surfers gave all stars to one blogger. Only O and X have 30 left.

Since A and B are full, one gives 30 to C, the other gives 30 to D.

Step 6: Balance Column C.

Total C = $C_{M} + C_{N} + C_{O} + C_{P} + C_{X} + C_{Y} = 45$ .

$C_{M} + C_{N} + 30 + 0 + 0 + 0 = 45 ⟹ C_{M} + C_{N} = 15$ .

Since $C_{M} \in {5, 15}$ and $C_{N} \in {0, 5}$ , the only valid sum is $C_{M} = 15, C_{N} = 0$ .

This means $D_{M} = 5, D_{N} = 5$ .

Step 7: Count recipients for each surfer.

M: A(10), C(15), D(5) $\to$ 3 bloggers.

N: A(25), D(5) $\to$ 2 bloggers.

O: C or D $\to$ 1 blogger.

P: A(5), B(25) $\to$ 2 bloggers.

X: C or D $\to$ 1 blogger.

Y: A(5), B(20), D(5) $\to$ 3 bloggers.

Surfers with exactly 2 bloggers: N and P. Total = 2.

Answer: 2

A question from this chapter

Question 1

Common Description: Instructions [29 - 34] Ten objects o1, o2, \dots, o10 were distributed among Amar, Barat, Charles, Disha, and Elise. Each item went to exactly one person. Each person got exactly two of the items, and this pair of objects is called her/his bundle. The following table shows how each person values each object. o1 o2 o3 o4 o5 o6 o7 o8 o9 o10 Amar 4993738795 Barat 59755368108 Charles 8883645896 Disha 8885536498 Elise 68956563710 The value of any bundle by a person is the sum of that person’s values of the objects in that bundle. A person X envies another person Y if X values Y’s bundle more than X’s own bundle. For example, hypothetically suppose Amar’s bundle consists of o1 and o2, and Barat’s bundle consists of o3 and o4. Then Amar values his own bundle at 4 + 9 = 13 and Barat’s bundle at 9 + 3 = 12. Hence Amar does not envy Barat. On the other hand, Barat values his own bundle at 7 + 5 = 12 and Amar’s bundle at 5 + 9 = 14. Hence Barat envies Amar. The following facts are known about the actual distribution of the objects among the five people. 1. If someone’s value for an object is 10, then she/he received that object. 2. Objects o1, o2, and o3 were given to three different people. 3. Objects o1 and o8 were given to different people. 4. Three people value their own bundles at 16. No one values her/his own bundle at a number higher than 16. 5. Disha values her own bundle at an odd number. All others value their own bundles at an even number. 6. Some people who value their own bundles less than 16 envy some other people who value their own bundle at 16. No one else envies others. What is Amar’s value for his own bundle?

Question 2

Common Description: Instructions [35 - 39] Adhara, Bithi, Chhaya, Dhanavi, Esther, and Fathima are the interviewers in a process that awards funding for new initiatives. Every interviewer individually interviews each of the candidates individually and awards a token only if she recommends funding. A token has a face value of 2, 3, 5, 7, 11, or 13. Each interviewer awards tokens of a single face value only. Once all six interviews are over for a candidate, the candidate receives a funding that is Rs.1000 times the product of the face values of all the tokens. For example, if a candidate has tokens with face values 2, 5, and 7, then they get a funding of Rs.1000 \times (2 \times 5 \times 7) = Rs.70,000. Pragnyaa, Qahira, Rasheeda, Smera, and Tantra were five candidates who received funding. The funds they received, in descending order, were Rs.390,000, Rs.210,000, Rs.165,000, Rs.77,000, and Rs.66,000. The following additional facts are known: 1. Fathima awarded tokens to everyone except Qahira, while Adhara awarded tokens to no one except Pragnyaa. 2. Rashida received the highest number of tokens that anyone received, but she did not receive one from Esther. 3. Bithi awarded a token to Smera but not to Qahira, while Dhanavi awarded a token to Qahira but not to Smera. How many tokens did Qahira receive?

Question 3

Common Description: Instructions [35 - 39] Adhara, Bithi, Chhaya, Dhanavi, Esther, and Fathima are the interviewers in a process that awards funding for new initiatives. Every interviewer individually interviews each of the candidates individually and awards a token only if she recommends funding. A token has a face value of 2, 3, 5, 7, 11, or 13. Each interviewer awards tokens of a single face value only. Once all six interviews are over for a candidate, the candidate receives a funding that is Rs.1000 times the product of the face values of all the tokens. For example, if a candidate has tokens with face values 2, 5, and 7, then they get a funding of Rs.1000 \times (2 \times 5 \times 7) = Rs.70,000. Pragnyaa, Qahira, Rasheeda, Smera, and Tantra were five candidates who received funding. The funds they received, in descending order, were Rs.390,000, Rs.210,000, Rs.165,000, Rs.77,000, and Rs.66,000. The following additional facts are known: 1. Fathima awarded tokens to everyone except Qahira, while Adhara awarded tokens to no one except Pragnyaa. 2. Rashida received the highest number of tokens that anyone received, but she did not receive one from Esther. 3. Bithi awarded a token to Smera but not to Qahira, while Dhanavi awarded a token to Qahira but not to Smera. How many tokens did Chhaya award?

Question 4

Common Description: Instructions [29 - 34] Ten objects o1, o2, \dots, o10 were distributed among Amar, Barat, Charles, Disha, and Elise. Each item went to exactly one person. Each person got exactly two of the items, and this pair of objects is called her/his bundle. The following table shows how each person values each object. o1 o2 o3 o4 o5 o6 o7 o8 o9 o10 Amar 4993738795 Barat 59755368108 Charles 8883645896 Disha 8885536498 Elise 68956563710 The value of any bundle by a person is the sum of that person’s values of the objects in that bundle. A person X envies another person Y if X values Y’s bundle more than X’s own bundle. For example, hypothetically suppose Amar’s bundle consists of o1 and o2, and Barat’s bundle consists of o3 and o4. Then Amar values his own bundle at 4 + 9 = 13 and Barat’s bundle at 9 + 3 = 12. Hence Amar does not envy Barat. On the other hand, Barat values his own bundle at 7 + 5 = 12 and Amar’s bundle at 5 + 9 = 14. Hence Barat envies Amar. The following facts are known about the actual distribution of the objects among the five people. 1. If someone’s value for an object is 10, then she/he received that object. 2. Objects o1, o2, and o3 were given to three different people. 3. Objects o1 and o8 were given to different people. 4. Three people value their own bundles at 16. No one values her/his own bundle at a number higher than 16. 5. Disha values her own bundle at an odd number. All others value their own bundles at an even number. 6. Some people who value their own bundles less than 16 envy some other people who value their own bundle at 16. No one else envies others. What BEST can be said about the distribution of object o1?

Question 5

Common Description: Instructions [29 - 32 ] Six web surfers M, N, O, P, X, and Y each had 30 stars which they distributed among four bloggers A, B, C, and D. The number of stars received by A and B from the six web surfers is shown in the figure below. Web surfer Stars received by A Stars received by B M 100 N 250 O 00 P 525 X 00 Y 520 The following additional facts are known regarding the number of stars received by the bloggers from the surfers. 1. The numbers of stars received by the bloggers from the surfers were all multiples of 5 (including 0). 2. The total numbers of stars received by the bloggers were the same. 3. Each blogger received a different number of stars from M. 4. Two surfers gave all their stars to a single blogger. 5. D received more stars than C from Y. How many surfers distributed their stars among exactly 2 bloggers?

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Allocation, Distribution and Selection Logic Previous Year Questions (PYQs) for CAT: 22+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Allocation, Distribution, and Selection Logic

The Core of Object Distribution

The Setup

The Matrix

The Goal

Allocation, Distribution and Selection Logic: Solved Questions with Step-by-Step Explanations (5 Problems)

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