12620256 different combinations of 31 balloons can be chosen
What are combinations?
Combinations are mathematical operations that count the variety of configurations that can be made from a set of objects, where the order of the selection is irrelevant. You can choose any combination of the things in any order.
From the question, we understand that; a combination of 31 is to be selected. Because the order is not important, we make use of combination.
Also, because repetition is allowed; different balloons of the same kind can be selected over and over again.
So:
n= 31 +8-1 = 38
r=31
Selection = [tex]^{38}C_{31}[/tex]
Use the formula:
[tex]^{n}C_{r}=\frac{n!}{(n-r)! r!}[/tex]
= > [tex]^{38}C_{31}=\frac{38!}{(38-31)! 31!}[/tex]
[tex]^{38}C_{31}=\frac{38!}{7! 31!}[/tex]
= [tex]\frac{38*37*36*35*34*33*32*31!}{7*6*5*4*3*2*1* 31!}[/tex]
= 12620256
To learn more about the combinations from the given link
https://brainly.com/question/26852614
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