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How many rational terms are present in the expansion of (1 + sqrt(2) + sqrt(5))⁶?

1. 7
2. 10
3. 6
4. 8

Correct answer: 10

Solution

In the multinomial expansion (1 + sqrt(2) + sqrt(5))⁶ = sum [6!/(a!b!c!)] * 1^a * (sqrt(2))^b * (sqrt(5))^c with a+b+c=6, a term is rational iff b and c are both even; the valid (b,c) pairs are (0,0),(0,2),(0,4),(0,6),(2,0),(2,2),(2,4),(4,0),(4,2),(6,0) — exactly 10 pairs.

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