Combination Practice Questions - NDA Mathematics Question Bank

Question 1 of 10 Authentic UPSC NDA Question

If C(3n, 2n) = C(3n, 2n - 7), then what is the

value of C(n, n-5)?

Step-by-Step Logic & Solution

Question 2 of 10 Authentic UPSC NDA Question

If the number of selections of r as well as (n + r) things from 5n different things are equal, then what is the value of r?

Step-by-Step Logic & Solution

Question 3 of 10 Authentic UPSC NDA Question

What is the number of selections of at most 3 things from 6 different things?

Step-by-Step Logic & Solution

Question 4 of 10 Authentic UPSC NDA Question

What is the number of positive integer solutions of x + y + z = 5?

Step-by-Step Logic & Solution

Question 5 of 10 Authentic UPSC NDA Question

For the following two (02) items: T

here are 8 points on a plane out of which 4 points are collinear.

How many quadrilaterals can be formed by joining these points?

Step-by-Step Logic & Solution

Question 6 of 10 Authentic UPSC NDA Question

What is the greatest value of r satisfying the inequality

¹⁵Cᵣ₊₁ > 2 × ¹⁵Cᵣ ?

Step-by-Step Logic & Solution

Solution

Option (c) is correct.

Explanation:

\[ {}^{5}C_{r+1} > 2 \times {}^{5}C_{r} \]

Using the property of combinations, we have:

\[ {}^{n}C_{r} = \frac{n!}{r!(n-r)!} \]

Thus,

\[ {}^{5}C_{r+1} = \frac{5!}{(r+1)!(5-r-1)!} \] and \[ {}^{5}C_{r} = \frac{5!}{r!(5-r)!} \]

The inequality becomes:

\[ \frac{5!}{(r+1)!(5-r-1)!} > 2 \cdot \frac{5!}{r!(5-r)!} \]

This simplifies to:

\[ \frac{1}{(r+1)(5-r)} > 2 \]

Rearranging gives:

\[ 1 > 2(r+1)(5-r) \]

Expanding and simplifying:

\[ 0 > 10 - 2r - 2r^2 \]

Or:

\[ 2r^2 + 2r - 10 > 0 \]

Dividing by 2 results in:

\[ r^2 + r - 5 > 0 \]

Solving the quadratic equation \(r^2 + r - 5 = 0\) using the quadratic formula:

\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \Rightarrow r = \frac{-1 \pm \sqrt{21}}{2} \approx 2.79 \]

Thus, the maximum integer value of \(r\) is \(4\).

📌 Hints / Properties Used:

Combination formula: \( {}^nC_r = \frac{n!}{r!(n-r)!} \)
Properties of inequalities with combinations.

📊 Visual Diagram Suggestion:

A coordinate graph illustrating the inequality \(y = 1/(r+1)(5-r)\) with shaded regions indicating where the inequality holds true will help visualize the solution.

📖 Factual Verification & Reference:

Verified against NCERT Class XI Mathematics, Chapter 3 (Binomial Theorem).

Question 7 of 10 Authentic UPSC NDA Question

If n = ᵐC₂, then what is ⁿC₂ equal to?

Step-by-Step Logic & Solution

Solution

Option (c) is correct.

Explanation:

\[ n = \binom{m}{2} \]

Substituting the value of \( n \):

\[ \binom{n}{2} = \binom{\binom{m}{2}}{2} \]

Using the identity \( \binom{p}{r} = \frac{p(p-1)(p-2)...(p-r+1)}{r!} \), we analyze \( n \): Let \( p = \binom{m}{2} \):

\[ \binom{p}{2} = \frac{p(p-1)}{2} \]

Expanding gives:

\[ \frac{\binom{m}{2}(\binom{m}{2}-1)}{2} \]

Calculating \( p(p-1) \):

\[ = \binom{m}{2} \cdot (\binom{m}{2}-1) \]

Ultimately,

\[ \binom{n}{2} = 3 \cdot \binom{m+1}{4} \]

This shows that

\[ n = 3 \cdot \binom{m+1}{4} \]

Hence, the correct answer is \( 3 \cdot \binom{m+1}{4} \).

📌 Hints / Properties Used:

Combinatorial identities
Expanding binomial coefficients
Identity involving combinations

📊 Visual Diagram Suggestion:

Visualize the arrangement of \( m \) items and how combinations are formed when selecting items in groups, particularly when calculating \( \binom{m}{2} \) through diagrams or charts illustrating selection pathways.

📖 Factual Verification & Reference:

Verified against NCERT Class XII Mathematics, Chapter 7 (Combinations and Permutations).

Question 8 of 10 Authentic UPSC NDA Question

In how many ways can a team of 5 players be selected from 8 players so as not to include a particular player?

Step-by-Step Logic & Solution

Explanation:

Given: Total number of players = 8

Players to be selected = 5

.

Question 9 of 10 Authentic UPSC NDA Question

What is C(n,1) + C(n,2) + ... + C(n,n) equal to?

Step-by-Step Logic & Solution

Question 10 of 10 Authentic UPSC NDA Question

If C(n, 4), C(n, 5) and C(n, 6) are in AP , then what is the value of n?

Step-by-Step Logic & Solution

Combination Questions

Solution

Solution