Question 1164352
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Thanks for stopping and asking this question.


Yes, the binomial coefficient    {{{C[100]^8}}}    is equal to    {{{100!/(8!*92!)}}}.


But nobody in healthy mind calculates it by calculating  100!  as the product of 100 integer factors.



            Yes,   100! = 1*2*3*4*. . . *100,


            but there is   92!  in the denominator    92! = 1*2*3*4*...*92.



So,  92 factors in the numerator and denominator  CANCEL  each other,  leaving  VERY  SIMPLE  formula

for the binomial coefficient    {{{C[100]^8}}} = {{{(100*99*98*97*96*95*94*93)/(1*2*3*4*5*6*7*8)}}},


and there is  NO  ANY  PROBLEM  to calculate it manually  (which people never do,  again),  or using  TECHNOLOGY  (a pocket calculator,
an  Excel spreadsheet,  an Excel compact standard formula,  or online free of charge calculators in numerous  Internet web-sites).


Finally,  we live in  XXI  century, and the year is  2020  now . . . 


It is my answer to your question.


Everything that I told you in this post,  relates to very basics of rational calculations,  and it is  ASSUMED 

that the person wanting calculate binomial coefficients,  is familiar with it . . . 



            Your final statement is  "Yet the problem is so easy to calculate by hand and still not calculator."


            The value of the binomial coefficient  is    {{{C[100]^8}}} = 186087894300,

            and you say it is so easy to calculate it by hand and still not with a calculator?


            It is very interesting to me - how you will do this challenge . . . .  Try it and then tell me . . . Ha-ha-ha.


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If you still have questions,  I will be more than happy answer them and to be in help  (!)