How many words can be formed from the word 'alpha'? The letter 'a' may be used twice but the other letters may only be used once.
There are no restrictions on whether or not they're real words, just combinations of letters. So, I need to add together:
$5$ letter words $+$ $4$ letter words $+$ $3$ letter words $+$ $2$ letter words $+$ $1$ letter words
There are, $(5!/2!)=60$, 5 letter words. 120 total but then we divide out by duplicate words because we have two a's.
$1$ letter words $=4$
The part I am having trouble with is the duplication of 'a'. I can't seem to figure out how to find $2,3,$ and $4$ letter words. For 2 letter words, there were so few examples I just wrote them out and saw that there were 13. So,
20 possibilities and then we need to subtract out the ones that contain 2 of p,l, and h, as well as the second aa.
I'm having a hard time believing my work for 2 letter words is accurate and attempting to do 3 and 4 letter words suggests that it isn't. Let me know if there is anything I can to do clear up what is being asked. Thanks in advanced.