Question on combinatorics

Question

Given solution of this question:

I think that number of 5 elements subsets with 7 should be C(n-1, 4) instead of C(n,4) as we have already fixed 7 so number of elements left will be n-1. Please check

Answer 1

20 (Ans)

set cardinality = n

Number of 5 element sets = nC5

Number of 5 element sets containing 7 =  (n-1)C4    $\because$  we have to take 7 & from

   remaining n-1 elements we have to take 4 elements}.

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now , given in Question

      $\frac{nC5}{4}$ =  (n-1)C4

   =>  $\frac{n!}{4 * (n-5)! * 5! } = \frac{(n-1)!}{ (n-5)! * 4! }$

         which gives n = 20 (Ans)