Question

How many zeroes will be there in the expansion of the expression

1¹×2²×3³×……..×(100)^100

Answer 1

May be 100...
...... . .

Answer 2

There will be 1300 zeroes in the expansion of the expression.

Given.

Expression = $5^{5}$×$10^{10}$×$15^{15}$× ... ×$100^{100}$.

To find,

number of zeroes,

Concept used,

• As, 10 = 2×5, number of zeros in the product will depend on number of 2s and 5s present in the product.
• We know that number of 5's are less than that of 2's, so number of zeroes in product will be decided by number of 5's in product.
• As, 5's power are less than that of 2's, number of zeroes will make equal to the power of 5's.

Power of 5 = $5^{5}$×$10^{10}$×$15^{15}$× ... ×$100^{100}$.

Sum of powers = [5+10+15+20+...+100] + [25+50+75+100]

Since, powers are in Arithmetic Progression,

sum of n terms = $\frac{n}{2}[2a + (n-1)d]$

n = 20

a = 5

d= 5

$\frac{20}{2}[2(5) + (20-1)5]+[250]\\ \\10[10+95]+[250]\\\\10[105]+[250]\\\\1050+250.$

1300.

Therefore, number of zeroes in the expansion will be 1300.