Question

Find the number of zeroes in:
${100}^1\times {99}^2\times {98}^3\times {97}^4\times \cdots 1^{100}$ ___

Answer 1

The key here is to count the number of fives. This can get done by:
${100}^1\times {95}^6\times {90}^{11}\times {85}^{16}\times {80}^{21}\times {75}^{26}\times \cdots 5^{96}$
(1 + 6 + 11 + 16 + 21 +26 + 31 +36 +41 +46+ . . . + 96) + (1 + 26 + 51 + 76)
$=20\times 48.5+4\times 38.5$
= 970 + 154 = 1124.