Find the number of zeros in the product:
1^1*2*2*3^3*4^4 ........... 98^98*99^99*100^100

Solution (By Examveda Team)

The number of zeroes in the answer will depend on two things.
1) on the number of powers of 10. For example 10^10 contains 10 zeroes.
2) On the number of 5s in the expression.( Because when five gets multiplied with 2 or 4 or any even number then the result will contain a zero in the units place..
SO finding the number of zeroes in powers of 10.
10^10 contains 10 zeroes.
10^20 contains 20 zeroes.
And son on
Hence the sum would be 10+20+30+40+.100=550.
But 100^100 can be written as (10*10)^100
Hence 100 zeroes are also to be counted. Hence number of zeroes from the powers of 10 comes out to be 550+100=650.
Now finding the number of 5s.
5^5 contains 5 fives.
15^15=(5*3)^15 contains 15 fives..
And so on..
Hence total number of fives will be 5+15+25+..+95=500.
But some fives are left uncounted..
They are..
25^25=(5*5)^25 and it had 25 fives left uncounted.
50^50=(5*10)^50 had 50 fives left uncounted.
75^75=(5*5*3)^75 had 75 fives left uncounted.
Hence the number of fives becomes 500 +15+50+75=650.
Hence the number of zeroes becomes 650+650=1300.