# Thread: Selections. Why am I short?

1. ## Selections. Why am I short?

Hi everyone,

This is a spin off from another question.

4 letters are chosen and repetition is allowed so there are $26^4= 456976$ selections possible.

Now I am going to split this up.

Letters all different = 26252423=358800
letters 2 the same= 26125244C2=93600
letters 3 the same=261125∗4C3=2600
Letters all the same=26

Add all these and I get 455026
WHY AM I SHORT???

Can anyone please tell me what selections I have missed out when I pulled apart the total?

2. ## Re: Selections. Why am I short?

You're missing combos with 2 pairs of letters

There are 26*25 ways to pick the letters and

$\dfrac 1 2 \begin{pmatrix}4 \\2\end{pmatrix}$ ways of arranging them in 4 slots.

The $\dfrac 1 2$ is because within the pair the letters can be flipped and you obtain the same code.

3. ## Re: Selections. Why am I short?

4 letters are chosen and repetition is allowed so there are 264=456976 selections possible.

Now I am going to split this up.

Letters all different = 26∗25∗24∗23=358800
letters 2 the same (and two different)= 26∗1∗25∗24∗4C2=93600
letters 2 pairs the same = 26*1*25*1*4C2/2 = 26*1*25*6/2=1950
letters 3 the same=26∗1∗1∗25∗4C3=2600
Letters all the same=26

358800+93600+1950+2600+26=456976.

That is excellent. Thank you Romsek!