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NDA 2023 Maths · Mathematics · Permutations and Combinations · medium

If binom nr=binom ns, then necessarily r=s. 2. The number of diagonals of a convex n-gon is binom n2.

Single correct option

Consider the following statements:
1. If (nr)=(ns)\binom nr=\binom ns, then necessarily r=sr=s.
2. The number of diagonals of a convex nn-gon is (n2)\binom n2.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (D) Neither 1 nor 2

Statement 1 is false: (nr)=(nn−r)\binom nr=\binom n{n-r}, so equality also holds when r+s=nr+s=n (for example (82)=(86)\binom82=\binom86). Statement 2 is false: (n2)\binom n2 counts all segments joining pairs of vertices, including the nn sides, so the diagonals number (n2)−n=n(n−3)2\binom n2-n=\dfrac{n(n-3)}2. Neither statement is correct.
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Original question written in the NDA 2023 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake