# Answer to Question #17072 in Combinatorics | Number Theory for navdeep sindhu

Question #17072

how many zero"s are there in tha result of 1!*2!*3!.......99!

give m a shortcut(or formula)to solve suvh kind of problem

give m a shortcut(or formula)to solve suvh kind of problem

Expert's answer

you get one additional zero in the result in 2 cases:

1. from every multiple of 10

2. from every pair of multiples of 2 and 5.

Which can be formulated as, actually, from every multiple of 5 you get one more zero.

Let us count all multiples of 5 in every factorial!

Factorials from 1 to 4 has none. +0 zeros

Factorials from 5 to 9 has one each. + 5 zeros

Factorials from 10 to 14 has 2 each. + 10 zeros

Factorials from 15 to 19 has 3 each. + 15 zeros

Factorials from 20 to 24 has 4 each. + 20 zeros

Factorials from 25 to 29 has 5 each. + 25 zeros

Factorials from 30 to 34 has 6 each. + 30 zeros

Factorials from 35 to 39 has 7 each. + 35 zeros

Factorials from 40 to 44 has 8 each. + 40 zeros

Factorials from 45 to 49 has 9 each. + 45 zeros

Factorials from 50 to 54 has 10 each. + 50 zeros

Factorials from 55 to 59 has 11 each. + 55 zeros

Factorials from 60 to 64 has 12 each. + 60 zeros

Factorials from 65 to 69 has 13 each. + 65 zeros

Factorials from 70 to 74 has 14 each. + 70 zeros

Factorials from 75 to 79 has 15 each. + 75 zeros

Factorials from 80 to 84 has 16 each. + 80 zeros

Factorials from 85 to 89 has 17 each. + 85 zeros

Factorials from 90 to 94 has 18 each. + 90 zeros

Factorials from 95 to 99 has 19 each. + 95 zeros

Total: 950 zeros in 1!*2!*3!.......99!

1. from every multiple of 10

2. from every pair of multiples of 2 and 5.

Which can be formulated as, actually, from every multiple of 5 you get one more zero.

Let us count all multiples of 5 in every factorial!

Factorials from 1 to 4 has none. +0 zeros

Factorials from 5 to 9 has one each. + 5 zeros

Factorials from 10 to 14 has 2 each. + 10 zeros

Factorials from 15 to 19 has 3 each. + 15 zeros

Factorials from 20 to 24 has 4 each. + 20 zeros

Factorials from 25 to 29 has 5 each. + 25 zeros

Factorials from 30 to 34 has 6 each. + 30 zeros

Factorials from 35 to 39 has 7 each. + 35 zeros

Factorials from 40 to 44 has 8 each. + 40 zeros

Factorials from 45 to 49 has 9 each. + 45 zeros

Factorials from 50 to 54 has 10 each. + 50 zeros

Factorials from 55 to 59 has 11 each. + 55 zeros

Factorials from 60 to 64 has 12 each. + 60 zeros

Factorials from 65 to 69 has 13 each. + 65 zeros

Factorials from 70 to 74 has 14 each. + 70 zeros

Factorials from 75 to 79 has 15 each. + 75 zeros

Factorials from 80 to 84 has 16 each. + 80 zeros

Factorials from 85 to 89 has 17 each. + 85 zeros

Factorials from 90 to 94 has 18 each. + 90 zeros

Factorials from 95 to 99 has 19 each. + 95 zeros

Total: 950 zeros in 1!*2!*3!.......99!

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