Originally Posted by

**Soroban** Hello, utopiaNow!

You're not considering WHO gets the pairs, etc.

__Score 0__

Four different numbers: $\displaystyle abcd$

Number of ways: .$\displaystyle 6\cdot5\cdot4\cdot3 \:=\:{\color{blue}360}$

__Score 1__

A pair and two singles: $\displaystyle aabc$

There are: .$\displaystyle {4\choose2} = 6$ distributions . . . $\displaystyle \{aabc, abac, abca, baac, baca, bcaa\}$

Number of ways: .$\displaystyle 6\cdot(6\cdot5\cdot4) \:=\:{\color{blue}720}$

__Score 2__

Two pairs: $\displaystyle aabb$

There are 3 distributions.

. . If the four players are $\displaystyle A,B,C,D$, they can be paired in 3 ways:

. . . . $\displaystyle \{AB,\,CD\},\;\{AC,\,BD\},\:\{AD,\,BC\}$

Number of ways: .$\displaystyle 3\cdot(6\cdot5) \;=\;{\color{blue}90}$

__Score 3__

A triple and a single: $\displaystyle aaab$

There are 4 choices of __who__ gets the triple.

Number of ways: .$\displaystyle 4\cdot(6\cdot5) \;=\;{\color{blue}120}$

__Score 4__

A quadruple: $\displaystyle aaaa$

The only choice is the value of the quadruple.

Number of ways: .$\displaystyle {\color{blue}6}$

Check: .$\displaystyle 360 + 720 + 90 + 120 + 6 \;=\;{\bf1296}$