Five letters are to be selected from the letters of the word 'ADVANTAGE'. How many different combinations (not permutations) are there? Have I got the correct answer and is there an easier way of achieving the correct solution? ![attempted_solution]( Thank you in advance

The word ADVANTAGE has $7$ distinct letters: $3$ A's, $1$ D's , $1$ V's , $1$ N's , $1$ T's , $1$ G's and $1$ E's .

So in how many ways can you exactly choose $5$ letters here?
Let us list them.

1. $5$ distinct letters
2. $3$ identical letters and $2$ distinct letters
3. $2$ identical letters and $3$ distinct letters



Hence, there are $3$ such ways.

Now for the calculation part: in $(1)$, we have ${7\choose 5}=21$ choices ; in $(2)$, we have ${1\choose 1}\cdot{6\choose 2}=15$ choices and in $(3)$, we have ${1\choose 1}\cdot{6\choose 3}=20$ choices.

> **_Note_** : The ${1\choose 1}$ comes from the fact there is only one triplet i.e. $3$ A's and only one duplet i.e. $2$ A's.

Total no. of choices = $\color{blue}{56}$.