CAT Permutations & combinations Questions

Counting problems distinguish arrangements, where order matters, from selections, where it does not. Restrictions, repeated objects and casework determine which counting method is appropriate.

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Key concepts

• Addition and multiplication principles
• Ordered arrangements versus selections
• Restrictions, repeated objects and complementary counting

CAT permutations & combinations practice questions

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Common mistakes

• Counting the same outcome in overlapping cases.
• Using a permutation formula for an unordered selection.

Practical solving approaches

• Define exactly what one outcome represents before counting.
• Use disjoint cases or count all outcomes and subtract the forbidden ones.

Permutations & combinations FAQs

How do I decide whether order matters?

Swap two selected objects. If the swap produces a different outcome in the problem, order matters; if not, you are counting a selection.

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CAT Permutations & combinations Questions

Counting problems distinguish arrangements, where order matters, from selections, where it does not. Restrictions, repeated objects and casework determine which counting method is appropriate.

16 free practice questions

Key concepts

• Addition and multiplication principles
• Ordered arrangements versus selections
• Restrictions, repeated objects and complementary counting

CAT permutations & combinations practice questions

Question 1Multiple choiceHARD

Question 1

Select the correct alternative from the given choices.

Choose the correct alternative.

Suppose you have a Currency, named Miso, in three denominations I Miso, 10 Misos and 50 Misos. In how many ways can you pay a bill of 107 Misos?

 

Choose an answer

A.18 B.15 C.19 D.17 E.16

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Question 2Multiple choiceHARD

Question 2

Select the correct alternative from the given choices.

Answer the questions independently of each other.

Let S be a set of positive integers such that every element n of S satisfies the conditions

a) 1000 < n < 1200 b) every digit in n is odd Then how many elements of S are divisible by 3?

Choose an answer

A.9B.10C.11D.12

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Question 3Multiple choiceHARD

Question 3

Select the correct alternative from the given choices.

Answer the questions independently of each other.

What is the smallest integer n for which any subset of {1, 2, 3, ... 20} of size n must contain at least two numbers that differ by 8?

Choose an answer

A.12B.13C.8D.16

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Question 4Multiple choiceHARD

Question 4

Select the correct alternative from the given choices.

Answer the questions independently of each other.

From a bunch of five keys, three particular keys have to be used to open a combination lock, such that the second key can be used only after the first and the third can be used only after the second. What are the chances of opening the lock in a random selection of three keys?

Choose an answer

A.

B.

C.

D.

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Question 5Multiple choiceMEDIUM

Question 5

Type in your answer in the input box provided below the question.

Answer the questions independently of each other.

In how many ways can you arrange the letters \'APTITUDE\' so as to form words that begin and end with a vowel?

Choose an answer

A.640B.1280C.4320D.8640

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Question 6Type in the answerMEDIUM

Question 6

Type in your answer in the input box provided below the question.

Answer the questions independently of each other.

Six countries send their Prime Ministers and Presidents for a UN conclave. In a round of six discussions, held at the conclave each discussion was between a Prime Minister and a President, both from different countries. Find the possible number of ways in which this conclave could have been conducted?

Your answer

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Question 7Type in the answerMEDIUM

Question 7

Type in your answer in the input box provided below the question.

Answer the questions independently of each other.

A box filled with more than 100 balls contains balls of blue, green, red and orange colour. In how many ways, Chakrabir can select 10 balls out of the box if it is known that the box has atleast 10 balls of each colour?

Your answer

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Question 8Type in the answerMEDIUM

Question 8

Type in your answer in the input box provided below the question.

Answer each of the questions independently of each other.

Each of the four servers are connected to eight other machines by two different lines each, while the four servers are connected to each other by three different lines (Note: the lines are direct). How many direct connections are there?

Your answer

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Question 9Type in the answerMEDIUM

Question 9

Type in your answer in the input box provided below the question.

Refer to the data below and answer the questions that follow.

A company is developing three-digit codes for their employees based on the following rules: a] The first digit (i.e., the leftmost digit) should be prime. b] The second digit should be even. c] If the second and first digits are same then the last digit should be odd. d] If the second digit is different from the first digit, the third digit should be different from the other two. Assume zero to be an even number.

Your answer

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Question 10Multiple choiceHARD

Question 10

Select the correct alternative from the given choices.

Answer the questions independently of each other.

For a positive integer n, let p_n denote the product of the digits of n, and s_n denote the sum of the digits of n. The number of integers between 10 and 1000 for which p_n + s_n = n is

Choose an answer

A.81B.16 C.18 D.9

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Question 11Multiple choiceHARD

Question 11

Select the correct alternative from the given choices.

Answer the following questions given below.

How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed?

Choose an answer

A.499B.500 C.375D.376 E.501

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Question 12Type in the answerMEDIUM

Question 12

Type in your answer in the input box provided below the question.

Answer the questions independently of each other.

How many three-digit numbers have at least one digit as 2 and at least one digit as 3?

Your answer

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Question 13Type in the answerEASY

Question 13

Type in your answer in the input box provided below the question.

Answer the questions independently of each other.

How many five digit numbers are there whose product of digits is 2000?

Your answer

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Question 14Multiple choiceMEDIUM

Question 14

Select the correct alternative from the given choices.

Answer the questions independently of each other.

In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. It was found that in 45 games both the players were girls and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is

Choose an answer

A.200B.216C.235D.256

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Question 15Multiple choiceMEDIUM

Question 15

Select the correct alternative from the given choices.

Answer the questions independently of each other.

There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

 

Choose an answer

A.5B.21C.33D.60

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Question 16Type in the answerHARD

Question 16

Type in your answer in the input box provided below the question.

Answer each of the questions independently of each other.

Rajesh wanted to organise his files and decided to use a code consisting of two distinct digits out of 0, 1, 2, 3, ... , 9 such that the first digit of the code is non zero. But when he stuck the coding on the files, he realised that the code 89 could be read as 68 when seen upside down. How many such numbers are there for which Rajesh may be confused when he reads them upside down?

Your answer

Check answerView solution

Common mistakes

• Counting the same outcome in overlapping cases.
• Using a permutation formula for an unordered selection.

Practical solving approaches

• Define exactly what one outcome represents before counting.
• Use disjoint cases or count all outcomes and subtract the forbidden ones.

Permutations & combinations FAQs

How do I decide whether order matters?

Swap two selected objects. If the swap produces a different outcome in the problem, order matters; if not, you are counting a selection.

More Modern Maths practice

• Probability →
• Set theory & Venn diagrams →