prob stats

Question

How many different? 10-letter words? (real or? imaginary) can be formed from the following? letters?
S, Y, E, N, A, R, Y, D, D, D

Answer 1

Notice that YY is the same as YY; it doesn't matter what order the two Y's appear in. Same for the three D's.

 

There are ten choices for where to put the S, nine remaining choices for where to put the first Y, eight remaining choices for where to put the E, etc. This gives

 

10*9*8*7*6*5*4*3*2*1 = 10!

 

Now back to the repeated letters. There are two = 2! arrangements of the two Y's, and 6 = 3! arrangements of the 3 D's. So we divide because those are the factors we've over-counted by:

 

10!/(2!*3!) = 302,400

 

This is based on the "multinomial theorem". What we just computed is written "10-choose-2,3". Note that all the other letters appear only once, so we can think of dividing by 1! for each of them, which doesn't change anything.