A person is permitted to select at least one and at most n coins from a collection of (2n+1) distinct coins. If the total number of ways in which he can select coins is 255, then n equals
A
4
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B
8
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C
16
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D
32
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Solution
The correct option is B4 We have 2n+1C1+2n+1C2+⋯+2n+1Cn=255 ...(i) Also, the sum of binomial coefficients 2n+1C0+2n+1C1+2n+1C2+⋯+2n+1Cn+2n+1Cn+1+2n+1Cn+2+⋯+2n+1C2n+1=(1+1)2n+1=22n+1 ⇒2n+1C0+2{2n+1C1+2n+1C2+⋯+2n+1Cn}+2n+1C2n+1=22n+1 ⇒1+2(255)+1=22n+1 ⇒1+255=22n ⇒22n=28 ⇒n=4