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Manipal MET2013MathematicsBinomial Theorem

A person is permitted to select at least one and at most n coins from a collection of (2 n+1) distinct coins. It the total number of ways in which he can select coins is 255 , then n equals

Options

  1. A4
  2. B8
  3. C16
  4. D32

Correct answer

A. 4

Step-by-step solution

We have ^ 2 n+1 C₁+ ^ 2 n+1 C₂+ + ^ 2 n+1 C_n=255 Also, the sum of binomial coefficients aligned & ^ 2 n+1 C₀+ ^ 2 n+1 C₁+ ^ 2 n+1 C₂+ + ^ 2 n+1 C_n & + 2 n+1 n+1 ++ 2 n+1 n+2 + + 2 n+1 2 n+1 aligned =(1+1)^ 2 n+1 =2^ 2 n+1 array r ^ 2 n+1 C₀+2 ^ 2 n+1 C₁+ ^ 2 n+1 C₂+ + ^ 2 n+1 C_n + 2 n+1 2 n+1 =2^ 2 n+1 array array rlrl & & 1+2(255)+1 & =2^ 2 n+1 & 1+255 & =2^ 2 n & & 2^ 2 n & =2^8 & & n & =4 array

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