Difference between revisions of "A choose b"

Revision as of 17:08, 15 June 2019

Here is the formula for a choose b: $\binom{a}{b}=\frac{a!}{b!(a-b)!}$ . This is assuming that of course $a \ge b$ .

Why is it important?

a choose b counts the number of ways you can pick b things from a set of a things. For example $\binom{8}{2}=\frac{8!}{2!(8-2)!}=\frac{8(7)}{2}=\frac{42}{2}=21$ . More at https://artofproblemsolving.com/videos/counting/chapter4/64.

a choose 2

Here is a list of n choose 2's

$\binom{2}{2}=1$

$\binom{3}{2}=3$

$\binom{4}{2}=6$

$\binom{5}{2}=10$

These are triangle numbers! My proof uses induction (assuming something is true unless proofed true or not true). $\binom{n}{2}=1+2+3...+(n-1)$ Then Simplify:

$\frac{n!}{2!(n-2)!}=\frac{n(n+1)}{2}$ More Simplify:

$\frac{n(n+1)}{2}=\frac{n(n+1)}{2}$

So now we have proved it. If you don't get what I did on the second step go to Proof Without Words on this wiki.

@@ Line 21: / Line 21: @@
 <math>\frac{n!}{2!(n-2)!}=\frac{n(n+1)}{2}</math> More Simplify:
-$\frac{n(n+1)}{2}=\frac{n(n+1)}{2}
+<math>\frac{n(n+1)}{2}=\frac{n(n+1)}{2}</math>
 So now we have proved it. If you don't get what I did on the second step go to Proof Without Words on this wiki.

Page

Toolbox

Search

Difference between revisions of "A choose b"

Revision as of 17:08, 15 June 2019

Why is it important?

a choose 2