If 35Cn + 7 = 35C4n - 2, then all the values of n are given

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If ³⁵C_{n + 7} = ³⁵C_{4n - 2}, then all the values of n are given by ?

1. 28
2. 3, 6
3. 3
4. 6

Correct Option: B

³⁵C_{n + 7} = ³⁵C_{4n - 2}
We know that if ⁿC_x = ⁿC_y then, Either x = y or x + y = n

Case 1: x + y = n
⇒ n + 7 + 4n - 2 = 35
⇒ 5n + 5 = 35
⇒ 5n = 30
∴ n = 6

Case 2: x = y
⇒ n + 7 = 4n - 2
⇒ 4n - n = 2 + 7
⇒ 3n = 9
∴ n = 3