**Problem:**

How many arrangements of 5 ‘s, 5 ‘s and ‘s are there with at least one and at least one between each successive pair of ‘s?

**Solution:**

There are three cases:

- Exactly one and one between each pair of ‘s: Between each of the four pairs of ‘s, the or the can be first — ways. The fifth and fifth along with the sequence of the rest of the letters can be considered as 3 objects to be arranged — ways. Altogether, ways.
- Exactly, one between each pair of ‘s and two ‘s between some pair of ‘s (or two ‘s between some pair of ‘s and exactly one between each pair of ‘s): there are four choices for between which pair of ‘s the two ‘s go and 3 ways to arrange the two ‘s and one there. There are two choices for whether the or the goes first between the other 3 pairs of ‘s and 2 choices for at which end of the arrangement the fifth goes. Multiplying by 2 for the case of two ‘s between some pair of ‘s, we obtain ways.
- Two ‘s between some pair of ‘s and two ‘s between some pair of ‘s. There are two subcases. If the two ‘s and two ‘s are between the same pair of ‘s, there are 4 choices for which pair of ‘s, ways to arrange them between this pair of ‘s, and choices for whether the or the goes first between the other 3 pairs of ‘s. If two ‘s and two ‘s are between the different pairs of ‘s, there are ways to pick between which ‘s the two ‘s and then between which ‘s the two ‘s go, ways to arrange the two ‘s and one and to arrange the one and two ‘s, and choices for whether the or the goes first between the other 2 pair of ‘s. Together, ways.

All together, the three cases give us a total of arrangements.

More later,

Nalin Pithwa

### Like this:

Like Loading...

*Related*