Question

What is the coefficient of the last term in the binomial expansion of (x + 1)⁹?

O 0
O 1
O 9
O 10

Answer 1

The coefficient of the last term in the binomial expansion of (x + 1)⁹ is 1, as the last term is defined by the binomial theorem and its binomial coefficients, which is always 1 for the last term of any binomial expansion.

Step-by-step explanation:

The coefficient of the last term in the binomial expansion of (x + 1)⁹ can be determined using the binomial theorem. The last term in the expansion is the one where all the n exponents have gone to the constant (1 in this case), which would be the term 1⁹. According to the binomial theorem, the general form of the expansion for (a + b)⁹ includes terms like a⁹n, 9a⁸b, down to ab⁸, and finally b⁹. The coefficient of b⁹ in our case is 1, because when expanding such a binomial, the coefficients are determined by the binomial coefficients, which, for the last term, are always 1 (there is 1 way to choose zero elements from n elements, hence the last term has a coefficient of 1).

Answer 2

In the binomial expansion of (x + 1)⁹, the coefficient of the last term is 1. This can be determined using the formula for the binomial coefficient in the expansion of (a + b)ⁿ, where the general term is given by C(n, k) * a^(n-k) * b^k. In this case, when k equals the exponent n, the binomial coefficient C(9, 9) is 1, and the expression becomes 1 * x^(9-9) * 1^9, which simplifies to 1. Therefore, the correct answer is 1.

The binomial expansion of (x + 1)⁹ is given by the formula:

(x + 1)^9 = Σ^9_(k=0) ⁹C(9, k) * x^(9-k) * 1^k

In this formula, C(9, k) represents the binomial coefficient, which is the number of ways to choose k elements from a set of 9. When k is equal to the exponent 9, the binomial coefficient C(9, 9) is 1. Therefore, the term with the highest power of x in the expansion is
`1*x^(9-9) *1^(9)` , which simplifies to 1. This means that the coefficient of the last term in the binomial expansion is 1, and the correct answer is 1. The other options ( 0, 9, 10) are not the coefficients of the last term in the expansion.