In the polynomial expansion of (x + y)^5, what is the coefficient of the x^2y^3 term?
A. 1
B. 4
C. 6
D. 10
E. 15
Answer (2)
To find the coefficient of the x 2 y 3 term in the expansion of ( x + y ) 5 , we can use the formula from the binomial theorem. The binomial theorem states that:
( x + y ) n = k = 0 ∑ n ( k n ) x n − k y k
where ( k n ) is a binomial coefficient calculated as:
( k n ) = k ! ( n − k )! n !
In this problem, we are given n = 5 and we want the term where x 2 y 3 . This means we need n − k = 2 for x 2 and k = 3 for y 3 .
Substituting these values into the binomial coefficient formula gives:
( 3 5 ) = 3 ! ⋅ ( 5 − 3 )! 5 ! = 3 ! ⋅ 2 ! 5 ⋅ 4 ⋅ 3 ! = 2 20 = 10
Therefore, the coefficient of the x 2 y 3 term is 10.
The correct answer is D. 10 .
Using the binomial theorem, the coefficient of the term x 2 y 3 in the expansion of ( x + y ) 5 is determined to be 10. This is calculated using the binomial coefficient ( 3 5 ) , which equals 10. Hence, the answer is D. 10.
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