AS level; binomial expansion

Question

Hi,

I am working on an AS maths question and am stuck. I would be very grateful with a little help. I have the answer, but do not understand the process.

The question is as follows:

"The coefficient of x² in a binomial expansion (1+x/2)ⁿ (where n is a positive integer) is 7.

Find the value of n."

The answer given is the following:

n(n-1)

-------- = 7

2! x 4

n(n-1) = 56

n2 -n -56 = 0

(n-8)(n+7) =0

n=8, -7

my understanding is that

ⁿC_r

= n!

---------

r!(n-r)!

so, I get the following:

ⁿC₂ = n! 1

--------- x ---

2!(n-2)! 2²

however, the book uses

n(n-1)

--------

2! x 4

How does

n(n-1) n!

-------- = ----------- ?

2! x 4 2! (n-2)! x 4

Answer 1

Martin, the right hand side has n! / (n-2)!

and that equals n(n-1)

n! = n (n-1) (n-2) (n-3) (n-4) ...

(n-2)! = (n-2) (n-3) (n-4) ...

As you see, all of (n-2)! Cancels out of top and bottom, leaving just n(n-1).