Question

Which of the following statements regarding the expansion of (x + y)^ n are correct ?

Answer 1

B. For any term (x^a)(y^b) in the expansion, a + b = n

Step-by-step explanation

The Binomial Theorem formula is,

`(a+b)^n=\sum_(k=0)^n\binom{n}{k}a^(n-k)b^k`

As we can see in this expression, the exponents of each term of the binomial are (n-k) and k, so, if we add them we have,

`n-k+k=n`

This means that the sum of the exponents is always n.

On the other hand, when k = 0, the exponents are n and 0, and the coefficient of that term is,

`\binom{n}{0}=(n!)/(0!(n-0)!)=(n!)/(n!)=1`

And, when k = n, the exponents are 0 and n, and the coefficient of that term is,

`\binom{n}{n}=(n!)/(n!(n-n)!)=(n!)/(n!0!)=(n!)/(n!)=1`

This means that for the first and last term (when the exponents for each variable are n) the coefficients are both 1.

Hence, the two true statements are B and C.