Re: No. of possible 'words'

Hello, Punch!

Quote:

In this question, a 'word' is defined to be any set of letters in a row,

whether of not it makes sense. . Find how many different 'words'

can be made using only 5 letters of the word SYLLABUS.

I tried to separate them into 4 cases,

1) all different letters

2) two S's

3) two L's

4) two S's and two L's

An excellent game plan! How far did you get?

We have 8 letters, 6 distinct: .$\displaystyle A,B,LL,SS,U,Y$

1) Five different letters

. . Choose 5 of the 6 letters and permutate them:.$\displaystyle _6P_5 \,=\,720$

2) Two S's: .$\displaystyle S\,S\,\_\,\_\,\_$

. . Choose 3 of the other 5 letters:.$\displaystyle _5C_3 \,=\,10$

. . and permute the 5 letters:.$\displaystyle 10\cdot\tfrac{5!}{2!} \,=\,600$

3) Two L's:.$\displaystyle L\,L\,\_\,\_\,\_$

. . Choose 3 of the other 5 letters:.$\displaystyle _5C_3\,=\,10$

. . and permutate the 5 letters:.$\displaystyle 10\cdot\tfrac{5!}{2!}\,=\,600$

4) Two S's, two L's:.$\displaystyle S\,S\,L\,L\,\_$

. . Choose 1 of the other 4 letters:.$\displaystyle _4C_1\,=\,4$

. . and permutate the 5 letters:.$\displaystyle 4\cdot\tfrac{5!}{2!\,2!}\,=\,120$

There are:.$\displaystyle 720 +600 + 600 +120 \:=\:2040$ "words".