NIMCET 2022 — Mathematics PYQ

Topic: Probability and Permutations

SEO Summary: Learn how to calculate the probability of a multi-digit number being divisible by 3 using combinatorics. This guide explains the divisibility rule for 3 and how to apply it to problems involving permutations without repetition.

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Question

A four-digit number is formed using the digits 1, 2, 3, 4, and 5 without repetition. The probability that it is divisible by 3 is:

A. $1/3$

B. $1/4$

C. $1/5$

D. $1/6$

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Solution

1. Total Number of Outcomes:

The total number of ways to form a four-digit number from 5 available digits $\{1,2,3,4,5\}$ without repetition is given by the permutation $P(5,4)$:

2. Divisibility Rule for 3:

A number is divisible by 3 if the sum of its digits is divisible by 3. We need to select 4 digits out of the 5 such that their sum is a multiple of 3.

The sum of all five digits is: $1+2+3+4+5=15$.

To find the sum of 4 digits, we subtract one digit ($x$) from the total sum (15). For the remaining sum to be divisible by 3, the digit $x$ must also be a multiple of 3.

• If we remove 3: The remaining digits are $\{1,2,4,5\}$. Sum $=1+2+4+5=12$ (Divisible by 3).

• If we remove any other digit (1, 2, 4, or 5), the sum will not be divisible by 3.

3. Favorable Outcomes:

There is only one set of four digits $\{1,2,4,5\}$ whose sum is divisible by 3.

The number of ways to arrange these 4 digits to form a four-digit number is $4!$:

4. Calculating Probability:

The probability $P$ is the ratio of favorable outcomes to total outcomes:

Correct Option:

C. $1/5$