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 $\displaystyle \binom{6}{1}$
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".
Already typed the following.
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 $\displaystyle \langle s_1,s_2,s_3,s_4\rangle$ where $\displaystyle 1\le s_i\le 6$, $\displaystyle i=1,\dots,6$. Here $\displaystyle s_i$ is the size of the selected ball of color $\displaystyle i$. Calculating the number of such tuples is easy taking into account that sizes can repeat.
If the order in which the balls were selected matters, then the previous result has to be multiplied by the number of permutations of 4 colors.