Expand (2x+5y)^6

Pascal triangle:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1<-

(2x+5y)^6=(2x)^6+6(2x)^5(5y)+15(2x)^4(5y)^2+20(2x)^3(5y)^3+15(2x)^2(5y)^4+6(2x)(5y)^5+
(5y)^6

The binomial coefficients for (a+b)^6 are:

1 6 15 20 15 6 1
which translates to:
(a+b)^6
=a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15 a^2b^4 + 6 ab^5 + b^6

Substitute a=2x, b=5y into the expansion to get the expansion of
(2x+5y)^6 accordingly.

i dont think the answers are correct pls give me another way our

i don't think the answer is correct, but please give me another way out

To expand the given expression, (2x + 5y)^6, we can use the binomial theorem, which states that for any two numbers a and b, and any positive integer n, the expansion of (a + b)^n can be written as the sum of the terms of the form C(n, r) * a^(n-r) * b^r, where C(n, r) is the binomial coefficient, given by the formula:

C(n, r) = n! / (r!(n-r)!)

In this case, a = 2x, b = 5y, and n = 6.

Now let's expand (2x + 5y)^6 using the binomial theorem:

(2x + 5y)^6
= C(6, 0) * (2x)^(6-0) * (5y)^0 + C(6, 1) * (2x)^(6-1) * (5y)^1 + C(6, 2) * (2x)^(6-2) * (5y)^2 + C(6, 3) * (2x)^(6-3) * (5y)^3 + C(6, 4) * (2x)^(6-4) * (5y)^4 + C(6, 5) * (2x)^(6-5) * (5y)^5 + C(6, 6) * (2x)^(6-6) * (5y)^6

Now, let's calculate each term individually:

Term 1:
C(6, 0) * (2x)^(6-0) * (5y)^0
= 1 * (2x)^6 * (5y)^0
= (2x)^6
= 64x^6

Term 2:
C(6, 1) * (2x)^(6-1) * (5y)^1
= 6 * (2x)^5 * (5y)^1
= 6 * 32x^5 * 5y
= 960x^5y

Term 3:
C(6, 2) * (2x)^(6-2) * (5y)^2
= 15 * (2x)^4 * (5y)^2
= 15 * 16x^4 * 25y^2
= 600x^4y^2

Term 4:
C(6, 3) * (2x)^(6-3) * (5y)^3
= 20 * (2x)^3 * (5y)^3
= 20 * 8x^3 * 125y^3
= 2000x^3y^3

Term 5:
C(6, 4) * (2x)^(6-4) * (5y)^4
= 15 * (2x)^2 * (5y)^4
= 15 * 4x^2 * 625y^4
= 3750x^2y^4

Term 6:
C(6, 5) * (2x)^(6-5) * (5y)^5
= 6 * (2x)^1 * (5y)^5
= 6 * 2x * 3125y^5
= 3750xy^5

Term 7:
C(6, 6) * (2x)^(6-6) * (5y)^6
= 1 * (2x)^0 * (5y)^6
= 1 * 1 * 15625y^6
= 15625y^6

Finally, add up all the terms:

(2x + 5y)^6 = 64x^6 + 960x^5y + 600x^4y^2 + 2000x^3y^3 + 3750x^2y^4 + 3750xy^5 + 15625y^6

So, the expansion of (2x + 5y)^6 is 64x^6 + 960x^5y + 600x^4y^2 + 2000x^3y^3 + 3750x^2y^4 + 3750xy^5 + 15625y^6.