# Permutation question: Pick out 2 items from 4 items which 2 duplicated, is there any formula?

If given ABBC, find the permutations if 2 letters are picked. If I calculate manually, they are: AB, AC, BA, BB, BC, CA, CB, 7 permutations. Is there any formula to solve this type of question? (Meaning, picking r number of letters from n number of letters with duplicated letters)

-
Depends a bit on what you mean by "this type of question". Are you always given 4 letters, or can that vary? Do they always include 3 distinct letters, or can that vary? Do you always pick 2 letters, or can that vary? – Gerry Myerson Aug 15 '11 at 12:38
For this type of question, I am meaning, picking r number letters from n number letters with k duplicated. – Allen Aug 15 '11 at 12:53

Suppose you had $n$ unique letters ${X_k}$, and letter $X_k$ were present in $r_k$ copies.

Then the number of unique pairs of letters, can be computed as $n (n-1) + \sum_{k} \mathrm{sgn} (r_k-1)$.

The term $n(n-1)$ count the number of pairs where letters are distinct, and remaining sum counts same letter pairs.

-
If $r_k\gt2$ then you want to count just one pair where both letters are $X_k$, not $r_k-1$ pairs. From AAA you just get AA, for example. – Gerry Myerson Aug 15 '11 at 12:34
@Gerry I realized this as well, I will edit as soon as I can. Thanks – Sasha Aug 15 '11 at 12:41
If picking 3 letters out of ABBC, is there any formula? – Allen Aug 15 '11 at 13:13
@Allen Suppose all $n$ letters are distinct. Then there are $n(n-1)(n-2)$ ways to pick triplets. Now if you have some distinct letters, you should subtract from that number the number of identical triples. – Sasha Aug 15 '11 at 13:17
@Sasha Thank you for your explanation. – Allen Aug 15 '11 at 14:25

Permutation of n things taken r at a time out of which k are repeated is

$\frac{n!}{(n-r)!k!}$

In your question , n=4 r=2, k=2

so total permutations is $\frac{4!}{2!2!}$=6

-
Allen has shown you 7 permutations, so if your formula gives 6, then something is wrong with your formula, or something is wrong with mathematics. – Gerry Myerson Aug 15 '11 at 12:36