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How many rational terms are present in the binomial expansion of (7^(1/7) + 11^(1/11))⁷⁷ ?

1. 2
2. 4
3. 6
4. 8

Correct answer: 2

Solution

The general term is C(77,r) * 7^((77-r)/7) * 11^(r/11). For this term to be rational, both 7^((77-r)/7) and 11^(r/11) must be rational, requiring (77-r)/7 and r/11 to both be non-negative integers. The first condition gives 7 | r (since 7 | 77), yielding r in {0,7,14,...,77} (12 values). The second gives 11 | r, yielding r in {0,11,22,...,77} (8 values). Both conditions require lcm(7,11) = 77 to divide r. In the range 0 to 77, only r = 0 and r = 77 satisfy this. Hence exactly 2 rational terms.

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