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If the number of integral terms in the expansion of (3^(1/2) + 5^(1/8))ⁿ is exactly 33, then the least value of n is-

1. 128
2. 264
3. 256
4. 248

Correct answer: 256

Solution

General term C(n,r)*3^((n-r)/2)*5^(r/8) is integral when r is a multiple of 8 and (n-r) is even. For all such r to give integers n must be even, then the count of valid r (r=0,8,...,n) is floor(n/8)+1. Setting this to 33 gives floor(n/8)=32, so the least n is 256.

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