Question

The coefficients corresponding to k = 0, 1, 2, ..., 5 in the expansion of (x + y)^5 are _____.

A. 1, 5, 10, 10, 5, 1
B. 1, 5, 10, 5, 1
C. 0, 1, 5, 10, 5, 1, 0
D. 0, 5, 10, 10, 5, 0

Answer 1

Answer: A. 1, 5, 10, 10, 5, 1

Explanation:

By the binomial expansion,

`(x+y)^n=\sum_(k=0)^(n) ^nC_k (x)^(n-k)(y)^k`

Where,

`^nC_k = (n!)/(k!(n-k)!)`

Here, n = 5 and k = 0,1,2,......, 5.

Thus,

`(x+y)^5=\sum_(k=0)^(5) ^5C_k (x)^(n-k)(y)^k`

`= ^5C_0 (x)^(5-0)y^0+^5C_1 (x)^(5-1)y^1+^5C_2 (x)^(5-2)y^2+^5C_3 (x)^(5-3)y^3+^5C_4 (x)^(5-4)y^4+^5C_5 (x)^(5-5)y^5`

`= 1 x^ 5 + 5 x^4 y ^1 + 10 x^3y^2+ 10 x^2y^3 + 5 x^1y^4 + 1 y^5` ( Because,
`a^0 = 1` )

Thus, the coefficients are 1, 5, 10, 5, 1.

⇒ Option A is correct.

Answer 2

The coefficients corresponding to k = 0, 1, 2, ..., 5 in the expansion of (x + y)^5 are 1, 5 , 10 , 10, 5, 1. The coefficients can be solve using manual expansion of the term of using combination:

nCr-1, where n is the power and r is the desired therm,