1. ## permutation

there are 24 balls of 6 different sizes in a bag there being 4 balls of each size in four different colours in how many ways 4 balls can be selected so that they are of different colours

2. Originally Posted by prasum
there are 24 balls of 6 different sizes in a bag there being 4 balls of each size in four different colours in how many ways 4 balls can be selected so that they are of different colours
The number of ways to select a particular color is $\binom{6}{1}$

The balls being different sizes is akin to having them the same size
but labelling them 1 to 6 so they are distinguishable.

Then, the number of sets of 4 different colored balls can be calculated.

The thread title suggests "arrangements" to be counted,
while the question asks for "selections".

Let us number sizes from 1 to 6 and colors from 1 to 4. If the order in which the balls are selected does not matter, then balls can be ordered by color after the selection. Therefore, a qualified selection can be identified with a 4-tuple $\langle s_1,s_2,s_3,s_4\rangle$ where $1\le s_i\le 6$, $i=1,\dots,6$. Here $s_i$ is the size of the selected ball of color $i$. Calculating the number of such tuples is easy taking into account that sizes can repeat.