Discussion

Explanation:

We know (x + y)n = nC0 × xnnC1 × xn-1 × y1 + nC2 × xn-2 × y2 + ... + nCn × yn

Let the three consecutive coefficients be nCr-1nCr & nCr+1

Now, nCr-1 nCr = 1 : 9
⇒ n!(r-1)!×(n-r+1)! : n!(r)!×(n-r)! = 1 : 9

⇒ rn-r+1 = 19

⇒ 9r = n - r + 1

⇒ 10r = n + 1   ...(1)

Also, nCnCr+1 = 9 : 63 = 1 : 7
⇒ n!(r)!×(n-r)! : n!(r+1)!×(n-r-1)! = 1 : 7

⇒ r+1n-r = 17

⇒ 7r + 7 = n - r

⇒ 8r = n - 7   ...(2)

Solving (1) & (2), we get

n = 39 & r = 4

Hence, 39.

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