How do I calculate combination 100C80?

Question

How do I calculate combination 100C80 when my calculator gives me error 2 and I need to get the answer fast during exam. is a binomial distribution question, but I'm stuck in calculating 100C80.

Answer 1

This happens when I'm extremely bored as I find an excuse not to study



100C80



= 100! / [ 80! 20!]



= (100 × 99 × 98 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 90 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 × 11 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19 × 20)



= (5 × 99 × 98 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 90 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 × 11 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (99 × 98 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 90 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(1 × 2 × 3 × 4 × 6 × 7 × 8 × 9 × 10 × 11 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (99 × 49 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 90 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(3 × 4 × 6 × 7 × 8 × 9 × 10 × 11 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (99 × 7 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 90 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(3 × 4 × 6 × 8 × 9 × 10 × 11 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (33 × 7 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 90 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(4 × 6 × 8 × 9 × 10 × 11 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (33 × 7 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(4 × 6 × 8 × 9 × 11 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (3 × 7 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(4 × 6 × 8 × 9 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (7 × 97 × 96 × 95 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(4 × 2 × 8 × 9 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (7 × 97 × 12 × 95 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(4 × 2 × 9 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (7 × 97 × 3 × 95 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(2 × 9 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (7 × 97 × 95 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 84 × 83 × 82 × 81)/(2 × 3 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (7 × 97 × 95 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 7 × 83 × 82 × 81)/(2 × 3 × 13 × 14 × 15 × 16 × 17 × 18 × 19)



= (7 × 97 × 5 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 7 × 83 × 82 × 81)/(2 × 3 × 13 × 14 × 15 × 16 × 17 × 18)



= (7 × 97 × 94 × 93 × 92 × 91 × 9 × 89 × 88 × 87 × 86 × 85 × 7 × 83 × 82 × 81)/(2 × 3 × 13 × 14 × 3 × 16 × 17 × 18)



= (7 × 97 × 94 × 93 × 92 × 7 × 9 × 89 × 88 × 87 × 86 × 85 × 7 × 83 × 82 × 81)/(2 × 3 × 14 × 3 × 16 × 17 × 18)



= (7 × 97 × 94 × 93 × 92 × 7 × 9 × 89 × 88 × 87 × 86 × 5 × 7 × 83 × 82 × 81)/(2 × 3 × 14 × 3 × 16 × 18)



= (7 × 97 × 47 × 93 × 92 × 7 × 9 × 89 × 88 × 87 × 86 × 5 × 7 × 83 × 82 × 81)/(3 × 14 × 3 × 16 × 18)



= (7 × 97 × 47 × 31 × 92 × 7 × 9 × 89 × 88 × 87 × 86 × 5 × 7 × 83 × 82 × 81)/(14 × 3 × 16 × 18)



= (7 × 97 × 47 × 31 × 92 × 7 × 3 × 89 × 88 × 87 × 86 × 5 × 7 × 83 × 82 × 81)/(14 × 16 × 18)



= (7 × 97 × 47 × 31 × 92 × 7 × 3 × 89 × 88 × 87 × 86 × 5 × 7 × 83 × 82 × 9)/(14 × 16 × 2)



= (7 × 97 × 47 × 31 × 46 × 7 × 3 × 89 × 88 × 87 × 86 × 5 × 7 × 83 × 82 × 9)/(14 × 16)



= (97 × 47 × 31 × 46 × 7 × 3 × 89 × 88 × 87 × 86 × 5 × 7 × 83 × 82 × 9)/(2 × 16)



= (97 × 47 × 31 × 23 × 7 × 3 × 89 × 88 × 87 × 86 × 5 × 7 × 83 × 82 × 9)/(16)



= (97 × 47 × 31 × 23 × 7 × 3 × 89 × 11 × 87 × 86 × 5 × 7 × 83 × 82 × 9)/(2)



= 97 × 47 × 31 × 23 × 7 × 3 × 89 × 11 × 87 × 43 × 5 × 7 × 83 × 82 × 9



= 97 × 47 × 31 × 23 × 7 × 3 × 89 × 11 × (3 × 29) × 43 × 5 × 7 × 83 × (2 × 41) × (3 × 3)



= 2 × 3⁴ × 5 × 7 × 11 × 23 × 29 × 31 × 41 × 43 × 47 × 83 × 89 × 97



= 10 × 7 × 11 × 23 × 29 × 31 × 41 × 43 × 47 × 81 × 83 × 89 × 97



= 770 × 23 × 29 × 31 × 41 × 43 × 47 × 81 × 83 × 89 × 97



= 770 × 661 × 31 × 41 × 43 × 47 × 81 × 83 × 89 × 97



= 770 × 661 × 31 × 1783 × 47 × 81 × 83 × 89 × 97



= and now I reach my limit...



= and here's the value from wolframalpha: 535983370403809682970



And if you don't like my answer, please TD this answer so I will feel sad to come back to Yahoo answers and answer anymore questions, and I will focus more on my studies. HAHAHAAHAHA



by the way, I never thought that you can solve it using a binomial distribution... will put this into my private list to look into it in the near future.

Answer 2

I don't know what an "error 2" is, but we did combinations before there were calculators that would handle them.



100 * 99 * 98 * 97 * 96 * 95 * 94 * 93 * 92 * 91 * 90 * 89 * 88 * 87 * 86 * 85 * 84 * 83 * 82 * 81

divide by

1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20

Answer 3

Combinations are calculated by using:



C(n,r) = n! / r!(n-r)!, where n= 100 and r = 80 then C(n,r) = 100! / 80!*(100-80)! = 5.359834 X 10^20 possible combinations.



The reason your professor choose such high numbers was to force you to use the expansion series since 100! will overload most calculators.



Hope this helps,



Newton1Law