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