Statistics Question?

Question

Determine the number of ways of selecting four letters from the word parallelogram, without disregard for the order. The answer is 150

Thanks

oops! Without disregard for the order, should read without REGARD for the order.

Answer 1

I hate these questions, yet I can't resist. Here's a brute force method, which considers all the various cases.

N = no. of ways

Case 1: 3 l's + one other
N = 7 ways of choosing last letter

Case 2: 3 a's + one other
N = 7

Case 3: 2 l's + 2 different letters
N = 7C2 = 21

Case 4: 2 a's + 2 different letters
N = 7C2 = 21

Case 5: 2 r's + 2 different letters
N = 7C2 = 21

Case 6: 2 l's + 2 a's; 2 l's + 2 r's; 2 a's + 2 r's
N = 3

Case 7: all different letters
N = 8C4 = 70

Add them up to get 150

Answer 2

you will have to break your problem here because there are 3 letters, which is a repeat: the three a's, the three l's, and 2 r's
Hint on the way to think of the problem:
the number of ways of selecting four letters, none of which is either r, l or a. that is 5C4 = 5
the number of ways of having exactly one of the repeated letters
3C1 * 5C3, {3C1: because there are three letters that have reps, l, a and r and the 5C3 because we need 3 more out of the remaining 5 letters}
the number of ways that we can have 2 of the repeated letters.
6* 5C2 {the repeats can be the same, as in rr, aa, or ll
or they can be different-la, ar, lr}
the number of ways of having 3 of the repeated letters
9* 5C1 {For the 9 factor: 3 identical-lll, aaa; 2 identical one different (count them) and 3 different: r, a, l- add them you get 9}
and the number of ways of having all letters from the repeated set.that is 10 ways.
the total is 150.