If coefficients of am and an in the expansion of (1 + a)m+n are α and β, then which one of the following is correct?a)α = 2βb)α = βc)2α = βd)α = (m + n) βCorrect answer is option 'B'. Can you explain this answer?

Question

If coefficients of am and an in the expansion of (1 + a)m+n are &alpha; and &beta;, then which one of the following is correct?a)&alpha; = 2&beta;b)&alpha; = &beta;c)2&alpha; = &beta;d)&alpha; = (m + n) &beta;Correct answer is option 'B'. Can you explain this answer?

Answer

The coefficients of am and an in the expansion of (1 + a)m n can be found using the binomial theorem.

The binomial theorem states that:

(1 + x)n = nCx xn

where nCx is the binomial coefficient, which is the number of ways to choose x items from a set of n items.

In our case, we want to find the coefficients of am and an in the expansion of (1 + a)m n. We can use the binomial theorem to expand the expression:

(1 + a)m n = nC0 (1)m a0 + nC1 (1)m-1 a1 + nC2 (1)m-2 a2 + ... + nCn (1)0 an

The coefficient of am is given by the term with a power of m, which is nCm (1)m a0. Similarly, the coefficient of an is given by the term with a power of n, which is nCn (1)0 an.

Therefore, the coefficients of am and an in the expansion of (1 + a)m n are nCm a0m and nCn an, respectively.

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