Question:

If $^nC_1 + 2\, ^nC_2 + .... + n\, ^nC_n = 2n^2$, then $ n = $

Updated On: May 11, 2024
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The Correct Option is A

Solution and Explanation

$^{n}C_{1}+2 ^{n}C_{2}+....+n ^{n}C_{n}=2n^{2}$
$ \Rightarrow \:\:\: \displaystyle\sum_{k=1}^{n} k \, ^nC_k = 2n^2$
$ \Rightarrow \:\:\: \displaystyle\sum_{k=1}^{n} \frac{k(n!)}{k!(n-k)!} = 2n^2$
$ \Rightarrow \:\:\: \displaystyle\sum_{k=1}^{n} \frac{k \, n(n-1)!}{k(k -1)!(n-k)!} = 2n^2$
$ \Rightarrow \:\:\: n\displaystyle\sum_{k=1}^{n} \frac{(n-1)!}{(k-1)!(n-k)!} = 2n^2$
$ \Rightarrow \:\:\: \displaystyle\sum_{k=1}^{n} \,^{n -1}C_{k - 1}= 2n$
$ \Rightarrow \:\:\: 2^{n -1} = 2n$
$n = 4$ satisfy the above equality
Hence, $n = 4$.
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Concepts Used:

Permutations and Combinations

Permutation:

Permutation is the method or the act of arranging members of a set into an order or a sequence. 

  • In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point. 
  • A permutation is used in many events of daily life. It is used for a list of data where the data order matters.

Combination:

Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.

  • Combination refers to the combination of about n things taken k at a time without any repetition.
  • The combination is used for a group of data where the order of data does not matter.