Combinations With Alike Objects

[dagr8est]

My book doesn't explain how to do combinations with alike objects but then it asks a whole page of questions on the topic. :?

How many 4-letter combinations are there of the letters in OUTLOOK? (Answer: 15)

I know how to do combinations with unique objects and all types of permutations but I have no idea how to do combinations with alike objects. Any help would be appreciated.

 

[pka]

I think that this is not the way your books intends you to work it.
But, \(\displaystyle \
{\rm{(1 + x)}}^{\rm{4}} (1 + x + x^2 + x^3 ) = 1 + 5x + 11x^2 + 15x^3 + {15}\limits_ \uparrow x^{4\limits^ \downarrow } + 11x^5 + 5x^6 + x^7\)
Note that the coefficient of x<SUP>4</SUP> is the answer.
There are the same number of 3-combinations. There are 11 5-combinations.
That is called the method of generating polynomials.

The ways I suspect you should do this is by noting a 4-conbination (which is really known as a selection) may contain no O’s, 1 O, 2 O’s or 3 O’s.

Then \(\displaystyle \L
\sum\limits_{k = 0}^3 {\left( {\matrix{
4 \cr
{4 - k} \cr

} } \right) = 15}\) . You are selecting 1, 2, 3, or 4 from {U,T,L,K}.