How do I solve this equation using Pascal's triangle? 4(3y+2)^5

Question

I need this answer pronto!

Answer 1

PASCAL'S Triangle

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

...

The 5th row of the triangle provides the coefficients in the expansion of (a+b)⁵.

Forget about the leading 4 for the moment, and make the substition of a=3y and b=2.

(a+b)⁵=a⁵+5a⁴b+10a³b²+10a²b³+5ab⁴+b⁵ becomes:

(3y+2)⁵=243y⁵+5(81y⁴×2)+10(27y³×4)+10(9y²×8)+5(3y×16)+32=

243y⁵+810y⁴+1080y³+720y²+240y+32.

Now multiply by 4: 972y⁵+3240y⁴+4320y³+2880y²+960y+128.

QUICK CHECK:

Add all these coefficients together and we get 12500=4(5⁵) when we plug in y=1.