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  • Find the number of terms free from radical sign in the expansion of <img alt="\left(1+3^{1 / 3}+7^{1 / 7}\right)^{10}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cleft%281&plus;3%5E

Find the number of terms free from radical sign in the expansion of \left(1+3^{1 / 3}+7^{1 / 7}\right)^{10}.

Option: 1

7


Option: 2

6


Option: 3

8


Option: 4

5


Answers (1)

best_answer

General term of the given expansion
=\frac{10 !}{\alpha ! \beta ! \gamma !} 1^\alpha \cdot 3^{\frac{\beta}{3}} \cdot 7^{\frac{\gamma}{7}} \text {, where } \alpha+\beta+\gamma=10\\ \therefore \text{Possible values of}\ \beta\ \text{are} : 0,3,6,9\\ \text{Possible values of}\ \gamma\ \text{are} : 0,7\\ \text{Possible values of}\ \alpha\ \text{are} : 0,1,2, \ldots, 10\\ Since, \alpha+\beta+\gamma=10\\ \therefore \text{ Possible triplets }\(\alpha, \beta, \gamma)\ are \(1,9,0),\\(4,6,0),(7,3,0), (10,0,0),(3,0,7),(0,3,7)\\ \therefore \quad \text{Number of terms free from radical sign} =6.

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