Maths · General term and middle term and simple applications
Number of irrational terms in the binomial expansion of ^{100} \) is
Number of irrational terms in the binomial expansion of \( \left(3^{1 / 5}+7^{1 / 3}\right)^{100} \) is
- A. 94
- B. 88
- C. 93
- D. 95
Step-by-step solution
The general term in the expansion is T_{r+1} = C(100, r) * 3^{(100-r)/5} * 7^{r/3}. A term is rational if both exponents are integers: (100-r) divisible by 5 and r divisible by 3. Let r=3k. Then 100-3k ≡ 0 mod 5 ⇒ 3k ≡ 0 mod 5 ⇒ k ≡ 0 mod 5. So k=5m, r=15m. With 0≤r≤100, m=0,1,...,6 gives 7 rational terms. Total terms=101, so irrational terms=101-7=94.
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