How can i find the 10th term of this binomial expansion?

Question

the tenth term (x^10) of (2x+y)^12

so what i did was:

(12 nCr 10) * (2x)^10 * (y)^2 =

66 * 1024x^10 * y^2 = 67,584x^10y^2

but apparently this is wrong cause the answer sheet gave me this:

1760x^3y^9

in my above answer, i chose 2 & 10 to be my exponents but the according to the answer, i was supposed to choose 3 & 9. how can one tell what to make as their exponents?

this is what i am really confused about :(

Answer 1

note: terms of (x+y) ^ k start (k nCr 0) x^k as 1st term., the 2nd is ( k Cr 1) x^(k-1) y....the 10th term is (k nCr 9) x^(k-9) y^9......{ (12 Cr 9) (2x)^3 y^9 }

Answer 2

The expansion of a binomial expressed as (X + Y)^n is:

(X + Y)^n = X^n + n(X^(n-1))Y/1! + n(n-1)(X^(n-2)Y^2/2! +...........Y^n

Note that the term number is equal to one less than the number subtracted from the exponent of X. Hence for for the tenth, the exponent of X must be (n- 9). Consider also that the expansion of a binomial raised to exponent n has (n +1) terms and that the coefficients of the first and last terms, second and second to the last term, third and third to the last terms etc are equal. Thus for the 10th term where n =12, the expression for the term would be:

(n)(n-1)(n -2)(X^(n-9)(Y^9)/3! = (12)(11)(10)((2X)^3)(Y^9)/3!

= 220(8)X^3Y^9 =( 1760)(X^3)(Y^9)