How many ways can a selection be done of 5 letters out of 5 A's, 4B's, 3C's, 2D's and 1 E.
Pick one
A) 60 B) 75 C) 71 D) 121 E) 221
We have the following letters: {AAAAA}, {BBBB}, {CCC}, {DD}, {E}   There are 7 different cases of 5 letter selections possible:   (5) - all letters are alike - 1 way, all A's;   (4, 1) - 4 letters are alike and 1 different - C12∗C14=8C21∗C41=8, where C12C21 is # of ways to choose which letter provides us with 4 letters from 2 (A or B) and C14C41 is # of ways to choose 5th letter from 4 letters left;   (3, 2) - 3 letters are alike and other 2 are also alike - C13∗C13=9C31∗C31=9, where C13C31 is # of ways to choose which letter provides us with 3 letters from 3 (A, B or C) and C13C31 is # of ways to choose which letter provides us with 2 letters from 3 (for example if we choose A for 3 letters then we can choose from B, C or D for 2 letters);   (3, 1, 1) - 3 letters are alike and other 2 are different - C13∗C24=18C31∗C42=18, where C13C31 is # of ways to choose which letter provides us with 3 letters from 3 (A, B or C) and C24C42 is # of ways to choose which 2 letters provides us with one letter each;   (2, 2, 1) - 2 letters are alike, another 2 letters are also alike and 1 is different - C24∗C13=18C42∗C31=18, where C24C42 is # of ways to choose which 2 letters provides us with 2 letters from 4 (A, B, C or D) and C13C31 is # of ways to choose which provides us with 5th letter from 3 letters left;   (2, 1, 1, 1) - 2 letters are alike and other 3 are different - - C14∗C34=16C41∗C43=16, where C14C41 is # of ways to choose which letter provides us with 2 letters from 4 (A, B, C or D) and C34C43 is # of ways to choose which 3 letters out of 4 provides us with one letter each;   (1, 1, 1, 1, 1) - all letters are distinct - 1 way (A, B, C, D, E).   Total:  1+8+9+18+18+16+1= 
<gadget id="calculator"> 1+8+9+18+18+16+1</gadget>
<output>71</output>
71.   Answer: 71.
<result>C</result>
How many ways can a selection be done of 5 letters out of 5 A's, 4B's, 3C's, 2D's and 1 E.