David Gwyn J. answered • 12/07/20
Highly Experienced Tutor (Oxbridge graduate and former tech CEO)
We want to find a specific term in the expanded expression (a + b2)10
There are 3 ways to do this:
(1) We could expand by multiplying out multiple times, but it would be quite time consuming! Maybe for powers of 2 or 3 you could consider it, but for 10 it's impractical.
OR
(2) We should use the Binomial Distribution formula which would look like this:
n
(a + b)n = ∑ ( ni ) (a) (n-i) b(i)
i=0
And for our expression a = a and b = b2 and n = 10
We want b4 term which is when i = 2 as (b2)2 = b4 hence the b term above is (b2)2
the a term is power of n-i which is 10-2 = 8 hence the a term above is (a)8
And, lastly, the coefficient (using factorials of a number!) of the term ( ni ) = ( 102 ) = n! / i! (n-i)! = 10! / 2! 8! = 10.9 /2 = 45
Hence our required term is 45a8b4
OR
(3) If you find it hard to remember the formula, you can make your own table of binomial coefficients (Pascal's Triangle), if you remember to start with 1 and 1 1 and add below as follows (which is also a bit cumbersome for powers of 10, but for smaller powers works quite well).
term
0--------------------------------------------1
1----------------------------------------1------1---
2---------------------------------1----------2---------1
3----------------------------1--------3-----------3---------1
4------------------------1--------4----------6----------4-------1
5---------------------1-----5--------10---------10--------5-----1
6------------------1----6-------15--------20--------15-------6----1
7---------------1----7-----21-------35--------35-------21------7---1
8-------------1---8----28-----56--------70-------56--------28---8---1
9-----------1---9---36----84-----126------126-----84-------36---9---1
10-------1-10-45---120---210----252------210----120----45--10---1
Which means the expanded expression looks like this:
a10 + 10a9b2 + 45a8b4 + 120a7b6 + 210a6b8 + 252a5b10 + 210a4b12 + 120a3b14 + 45a2b16 + 10ab18 + b20
And our required term is 45a8b4
