Question

The ratio of three consecutive binomial coefficients in the expansion of (1 + x)" is 5: 12: 20 find n

(1) 120 (2) 34 (3) 118 (4) 35

Answer 1

REFER TO THE ATTACHMENT!

THIS IS THE ANS ACCORDING TO GIVEN RATIO

HOPE IT HELPS :)

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Answer 2

The ratio of three consecutive binomial coefficient in the expansion of (1 + x)ⁿ is 2 : 5 : 12 ( given ratio in question is incorrect. ratio should be 2 : 5 : 12)

we have to find the value of n

solution : let three consecutive terms are k - 1, k and k + 1

so, coefficient of (k - 1)th term = $^nC_{k-1}$

coefficient of kth term = $^nC_k$

coefficient of (k + 1)th term = $^nC_{k+1}$

now, $^nC_{k-1}$ : $^nC_{k}$ : $^nC_{k+1}$ = 2 : 5 : 12

so, $\frac{^nC_{k-1}}{^nC_{k}}$ = 2/5

⇒k!(n - k)!/(n - k + 1)!(k - 1)! = 2/5

⇒k/(n - k + 1) = 2/5

⇒5k = 2n - 2k + 2

⇒7k = 2n + 2 ........(1)

again, $\frac{^nC_{k+1}}{^nC_{k}}$ = 12/5

⇒{1/(k + 1)!(n - k - 1)!}/{1/k!(n - k)!} = 12/5

⇒k!(n - k)!/(k + 1)! (n - k - 1)! = 12/5

⇒(n - k)/(k + 1) = 12/5

⇒5n - 5k = 12k + 12

⇒5n = 17k + 12

from equation (1) we get,

⇒5n = 17(2n + 2)/7 + 12

⇒35n = 34n + 34 + 84

⇒n = 118

Therefore the value of n = 118 , option (3) is correct choice.