NIMCET Permutations and Combinations Solved PYQs & Practice Sets

Permutations and Combinations (P&C) is a core topic in the NIMCET Mathematics section, directly yielding 3–5 questions (36–60 marks out of 600) every year. Furthermore, mastering P&C is essential because it forms the mathematical foundation for Probability, the single highest-weightage topic in NIMCET.

In this practice guide, we present authentic NIMCET past-year questions (PYQs) alongside fully worked solutions, shortcut applications, and a curated practice set.

1. Solved NIMCET Past-Year Questions (PYQs)

Question 1 (NIMCET PYQ – Selection & Formation)

In how many ways can a committee of 5 members be formed from 6 men and 4 women such that the committee contains at least 3 men?

Options

• (A) 186
• (B) 196
• (C) 210
• (D) 246

Solution

We need a committee of 5 members from 6 Men (M) and 4 Women (W) with at least 3 men.

Possible cases:

1. Case 1: 3 Men and 2 Women

   $\text{Ways} = \binom{6}{3} \times \binom{4}{2} = 20 \times 6 = 120$

2. Case 2: 4 Men and 1 Woman

   $\text{Ways} = \binom{6}{4} \times \binom{4}{1} = 15 \times 4 = 60$

3. Case 3: 5 Men and 0 Women

   $\text{Ways} = \binom{6}{5} \times \binom{4}{0} = 6 \times 1 = 6$

$\text{Total Ways} = 120 + 60 + 6 = \mathbf{186}$

Correct Answer: (A)

Question 2 (NIMCET PYQ – String & Gap Method)

How many words can be formed using the letters of the word "TRIANGLE" such that the vowels never come together?

Options

• (A) 36,000
• (B) 14,400
• (C) 21,600
• (D) 30,240

Solution

Word: TRIANGLE (8 distinct letters)

• Vowels: A, E, I (3 vowels)
• Consonants: T, R, N, G, L (5 consonants)

We apply the Complementary Method:

$\text{Vowels Never Together} = \text{Total Words} - \text{Vowels Always Together}$

1. Total unrestricted words

   $8! = 40,320$

2. Vowels always together

   Treat A, E, I as one block.

   • Block + 5 consonants = 6 units

     $6!$

   • Internal arrangements of vowels

     $3!$

   Therefore,

   $\text{Vowels Always Together} = 6! \times 3! = 720 \times 6 = 4,320$

3. Vowels never together

   $\text{Ways} = 40,320 - 4,320 = \mathbf{36,000}$

Correct Answer: (A)

Question 3 (NIMCET PYQ – Distribution of Identical Objects)

Find the number of non-negative integer solutions to the equation $x + y + z + w = 15$.

Options

• (A) 816
• (B) 456
• (C) 680
• (D) 969

Solution

This is a classic Stars and Bars (Beggar’s Method) problem.

The number of non-negative integer solutions to

$x_1 + x_2 + \dots + x_r = n$

is

$\binom{n+r-1}{r-1}$

Here, $n=15$ and $r=4$.

$\text{Solutions} = \binom{15+4-1}{4-1} = \binom{18}{3}$

$= \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = 816$

Correct Answer: (A)

2. High-Yield Practice Problem Set for NIMCET

Test your speed and concept clarity with these exam-level practice questions.

Practice Question 1

How many 3-digit numbers between 100 and 1000 can be formed using digits $0,1,2,3,4,5,6$ if no digit is repeated?

Solution

• Hundreds place: any digit except 0 → 6 choices
• Tens place: any remaining digit (including 0) → 6 choices
• Units place: any remaining digit → 5 choices

$\text{Total} = 6 \times 6 \times 5 = \mathbf{180}$

Practice Question 2

Out of 10 points in a plane, 4 points are collinear. How many distinct triangles can be formed?

Solution

• Total triangles from 10 points

  $\binom{10}{3} = 120$

• Invalid triangles from 4 collinear points

  $\binom{4}{3} = 4$

Therefore,

$\text{Actual Triangles} = 120 - 4 = \mathbf{116}$

3. Key Concepts Checklist for NIMCET P&C

• Fundamental Principles of Counting: Addition rule (OR cases) vs Multiplication rule (AND cases)
• Circular Permutations: Table arrangements $(n-1)!$; necklace arrangements $\frac{(n-1)!}{2}$
• Gap Method vs String Method: Gap Method when items must never be together; String Method when items must always be together
• Stars and Bars: Non-negative solutions $\binom{n+r-1}{r-1}$; positive solutions $\binom{n-1}{r-1}$

Frequently Asked Questions (FAQ)

Q1. How many questions are asked from P&C in NIMCET?

Approximately 3–5 questions every year, worth 36–60 marks.

Q2. Why is P&C important for other NIMCET topics?

P&C is the foundation for Probability, which carries the highest weightage in NIMCET Mathematics.

Q3. What is the formula for positive integer solutions?

For $x_i \ge 1$:

$\binom{n-1}{r-1}$

For $x_i \ge 0$:

$\binom{n+r-1}{r-1}$

Q4. What is the Gap Method?

Arrange the unrestricted items first, creating gaps, and then place the restricted items into those gaps so that they never come together.