binomial theorem help please!!?

Question

please help and explain!? i dont get this!!!


What is the coefficient of a^4b^6 in the expansion of (a + b)^10?

Answer 1

If you were to expand this out, you would get:

10C0 a^10 b^0 + 10C1 a^9 b^1 + 10C2 a^8 b^2... and so on

Do you see a pattern with the exponents for a and b?

The exponent for "b" is always the number to the right of the "C"

The exponent for "a" is always the difference between the two numbers beside the "C"

So, for a^4b^6

I know that the second number beside the "C" is going to be 6. So lets put that down first.

__C6

The missing number is going to be 10 because it is our total.

So your answer should be:

10C6 = 210

Answer 2

I don't remember the name of the triangle, but for you it would like this:

1
1 1
1 2 1
1 3 3 1
and it works like (a+b)^0=1
(a+b)^1=1a+1b
The triangle works by adding the number above the current line, when you get to (a+b)^10 the line is :
1 11 55 165 330. . . .
so it goes a^10 + 11a^9b + 55a^8b^2. . . the answer to your question is 330

Answer 3

The binomial theorem, for n natural number,

(a + b)^n = a^n + n a^(n-1) b + n (n-1) a^(n-2) b^2 / 2 + ... + (n! / ((n - i)! i!) a^(n - i) b^i + ... + b^n.

(a + b)^10=
a^10 + 10a^9b + 10.9.a^8.b^2 /2 + 10.9.8.a^7.b^3/(2.3) +

10.9..8.7.a^6.b^4/(2.3.4) + 10..9.8.7.6.a^5.b^5/(2.3.4.5) +

10.9. 8.7.6.5.a^4.b^6/( 2.3.4.5.6 ) + ........

The coefficient of a^4b^6 = (10.9. 8.7.6.5) / ( 2.3.4.5.6 )