Let's say, I have 4 yellow and 5 blue balls.

Erick Clay

Erick Clay

Answered question

2022-05-22

Let's say, I have 4 yellow and 5 blue balls. How do I calculate in how many different orders I can place them? And what if I also have 3 red balls?

Answer & Explanation

concludirgt

concludirgt

Beginner2022-05-23Added 11 answers

This is a standard problem involving the combinations of sets, though perhaps not very obvious intuitively.
Firstly consider the number of ways you can rearrange the entire set of balls, counting each ball as indepndent (effectively ignoring colours for now). This is simply ( 4 + 5 ) ! = 9 !, since the 1st ball can be any of the 9, the 2nd can be any of the remaining 8, and so on.
Then we calculate how many different ways the yellow balls can be arranged within themselves, since for the purpose of this problem they are considered equivalent. The number of combinations is of course 4 !; similarly, for the blue balls the number is 5 !.
Hence, overall we find:
total arrangements = arrangements of all balls arrangements of yellow balls × arrangements of blue balls
Therefore in our case we have:
total arrangements = 9 ! 5 ! × 4 ! = 126

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