When wo1uld you use the permutation formula as opposed to just x! ?

Answer 1

This link will give you a good start.

Answer 2

I'm guessing that you mean

c!/(r!*(c-r)!)

where c is some number of 'things' and r is the number of 'things' that you are going to 'take' from the set of c 'things'

For example, if you had a bag of marbles with a green, blue, red, and yellow marble in it, how many different ways can you take 1 marble out of it?

Pretty obviously, 4 ways. You could take a G, B, R, or Y marble.

Now 4!/(1!*(4-1)!) = 24/(1*6) = 4

How many ways can you take 2 marbles out?

(G,B) (G,R) (G,Y) (B,R) (BY) or (R,Y). That's 6 ways

4!/(2!*(4-2)!) = 24/(2*2) = 24/4 = 6

How about taking 3 at a time? I claim there are

4!/(3!*(4-3)!) = 24/(6*1) = 4 ways. And you could grab

(G,B,R) (G,B,Y) (G,R,Y) or (B,R,Y) and that's it. Four ways.

There is also a very kewl way to write this called 'Pascals Triangle' (I hope the positioning doesn't get screwed up in posting it) Yeah, it did get screwed up. Just imagine that it's a nice 'symetric' isoceles triangle

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

See how each number in each row is formed by adding together the number in the row just above and to the left and right of it? We number the rows from the top, starting with 0 (this is important) and, going from left to right, the entries are also numbered starting at 0 (also important).

Look close. If you go to the 4'the row and look across it, you see 1, 4, 6, 4, 1 and the 4,6,4 are exactly the numbers we came up with for the ways to choose 1, 2, or 3 marbles. What do the 1's on each end mean? The one on the right is in the 4'th place (we started counting at 0, remember?) and that's the number of ways you can take 4 marbles from a sack of 4. (Grab 'em all!) And the 1 on the left (the 0 place) is the number of ways you can take zero marbles. (Keep yer sticky fingers outta my marble bag!)

The row corresponds to the c in the equation, and the position number is the r. So...... How many ways can you take 2 marbles out of a bag of 5 marbled (Say G, B, R, Y, W for green, blue, red, yellow, and white)?

I claim it's 10, and I'll let you work out what the combinations are (remember that order doesn't count. Green and blue is the same as blue and green)

These numbers are also the coefficients of the polynomial
expansion of (a+b)ⁿ

It's called the 'binomial theorem', but that's a story for another day.


Doug