Arranging Letters and Plus Signs

Question

We have 4 spaces which should be filled with $1$ letter and $3$ plus signs, $2$ letters and $2$ plus signs, or with $3$ letters and $1$ plus sign. In any of these cases, letters can not be repeated and their orders and positions matter, and the plus signs are identical so their order does not matter.

Examples of valid arrangements:

++CY, +AT+, +++S, WXY+

Examples of invalid arrangements:

+++SA (we have exactly 4 spaces), U+JU (letters can not be repeated), AZQC (not satisfying any of the three given cases, note that the 4 spaces is neither letters only nor plus signs only).

Note: A+B+ and A++B are different since the positions of the letters matter.

How many different arrangements are there?

Clearly, for the first scenario (using $1$ letter and $3$ plus signs) we have $26\times4=104$ different arrangements. I am confused with the second and the third scenarios.

Answer 1

For the second scenario, think of it this way:

• First, you choose the $2$ positions that will contain the letters
• Then, you arrange two letters on the two chosen positions.