How many different/unique $4$-letter arrangements are there of the letters in the word Mississauga?

M=1

I=2

S=4

A=2

U=1

G=1

Case 1: All 4 letters are same

There is only one arrangement for this.

Case 2: 3 letters are similar

The three letters must be S. The remaining letter can be chosen in ${5 \choose 1}$ ways and these can be arranged in $\frac {4!}{3!}$ ways. This makes a total of 20 arrangements.

Case 3: 2 pairs are similar

The two repeating letters can be chosen in ${3 \choose 2}$ ways and can be arranged in $\frac {4!}{{2!}{2!}}$. This equals 18 arrangements.

Case 4: 2 are similar

The repeating letters can be chosen in ${3 \choose 1}$ ways and the remaining two in ${5 \choose 2}$ ways. These can be arranged in $\frac {4!}{2!}$ ways. This equals 360 arrangements.

Case 5: All 4 are different The 4 letters can be chosen in ${6 \choose 4}$ ways and can be arranged in $4!$ ways. This equals 360 ways.

Hence the total number of arrangements is $1+20+18+360+360=759$.

Here's a hint as to one way to do it. Break it up into mutually-distinct cases:

• All four letters are different.
• One letter appears twice; two letters appear once.
• Two letters appear twice.
• One letter appears three times; one letter appears once.
• One letter appears four times.

Can you take it from here?