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MATH 135 Winter 2009

Lecture VIII Notes

Binomial Theorem

Recall that we are trying to come up with a way of expanding (a+b)nwithout actually having to

expand it for each value of nin which we are interested. This is similar to wanting to come up with

“closed form” expressions for things like 12+ 22+· · · +n2.

Last time we introduced the notation n

r=n!

r!(n−r)! and did a few calculations.

Binomial Theorem (Theorem 4.34)

If aand bare any numbers and n∈P, then

(a+b)n=n

0an+n

1an−1b+· · · +n

ran−rbr+· · · +n

n−1abn−1+n

nbn

Alternatively, we can write (a+b)n=

n

X

r=0 n

ran−rbr.

We will prove this and do some calculations, but need to do look at a couple of preliminary re-

sults ﬁrst.

Proposition 4.33

If nand rare integers with 0 ≤r≤n, then n

ris an integer.

Rationale

We will not formally prove this. However, last time we looked at n

ras the number of ways of

choosing robjects from among nobjects. Since this number of ways is an integer, then n

rshould

be an integer.

Proposition 4.32

If nand rare integers with 1 ≤r≤n, then n+ 1

r=n

r+n

r−1.

Aside

It is quite likely that you have seen this Proposition before in the following picture:

1

1 1

121

1331

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

What’s this called? Can you see how this relates to Proposition 4.32?