2C-4-n-35C-3-n-2-n-

Question Number 74129 by malwaan last updated on 19/Nov/19

2 \boldsymbol C_{4}^{\boldsymbol n} = 35 \boldsymbol C_{3}^{\frac{\boldsymbol n}{2}}

\Rightarrow \boldsymbol n = ?

Answered by MJS last updated on 20/Nov/19

n \in N \Rightarrow n ⩾ 4 \land \frac{n}{2} ⩾ 3 \Rightarrow n ⩾ 6

n = 2 k (k ⩾ 3)

2 C_{4}^{2 k} = 35 C_{3}^{k}

2 \frac{(2 k)!}{4! (2 k - 4)!} = 35 \frac{k!}{3! (k - 3)!}

\frac{1}{3} k (k - 1) (2 k - 3) (2 k - 1) = \frac{35}{6} k (k - 2) (k - 1)

k = 0 \lor k = 1 not valid

\frac{1}{3} (2 k - 3) (2 k - 1) = \frac{35}{6} (k - 2)

2 (2 k - 3) (2 k - 1) = 35 (k - 2)

8 k^{2} - 51 k + 76 = 0

(k - 4) (8 k - 19) = 0

k = \frac{19}{8} not valid

k = 4

\Rightarrow n = 8

Commented by malwaan last updated on 20/Nov/19

t h a n k y o u s o m u c h s i r M J S

t h a n k y o u s o m u c h s i r M J S

Commented by MJS last updated on 20/Nov/19

{you}^{'} re welcome

{you}^{'} re welcome

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