Calculate all possible combinations

archidat

New member
Joined
Jan 15, 2021
Messages
2
I need help in solving the following problem. Three numbers are given: 6, 7, and 8. I need to calculate number of all possible combinations of 9-digit number consisting of 6,7 and 8 which satisfy the following condition: Quantity of 6- four, of 7-three, and of 8-two. So, I need calculate combinations such as 666677788, 687766678, 787678666, etc.

What I did: For a start. I seem to calculate the combinations with four 6s and the remaining combinations of 7 and 8, although I am not 100% sure:
9C4*(2^5). This should give the numbers with four 6s, with quantity of 7s and 8s from 1 to 5. From here, I do not know how to exclude those combinations that do not satisfy the given condition.
 
Last edited:
Place your 4 6's. There are [MATH]\dbinom{9}{4}[/MATH] ways of placing them. You now have 5 slots open.
Place your 7's. There are [MATH]\dbinom{5}{3}[/MATH] ways of placing these.
The 8's positions are now determined.

Total number of combos is [MATH]n=\dbinom{9}{4}\dbinom{5}{3}= 1260[/MATH]
 
So you want to know how many ways can you write down the 9 digits 6 6 6 6 7 7 7 8 8.

This is a standard counting problem! The answer is 9!4!3!2!\dfrac {9!}{4!3!2!}
Unfortunately for me, I am not good in counting so even standard questions seem hard to me.
 
Top