Question

What is the number of digits in the product $\left( 100-1 \right)\left( 100-2 \right)\left( 100-3 \right)....\left( 100-199 \right)\left( 100-200 \right)$?
A. $1$
B. $400$
C. Cannot be determined
D. None of the above

Answer 1

Hint: In this problem we need to calculate the number of digits in the product of the given equation. So, we need to first calculate the product of the given equation or expression. We can observe that the first term in the given expression is $100-1$ and the last term in the expression is $100-200$. In between the numbers are forming an arithmetic progression with a common difference of $-1$. Here we can observe the term $100-100$ which comes in the middle of the expression we can calculate this value also and find the required result.

Complete step by step solution:
Given expression is $\left( 100-1 \right)\left( 100-2 \right)\left( 100-3 \right)....\left( 100-199 \right)\left( 100-200 \right)$.
Considering the first term of the given expression which is $100-1$. We know that the value of $100-1$ is $99$. Considering the last term of the given expression which is $100-200$. We know that the value of $100-200$ is $-100$. So, the series forms arithmetic progression starting with $99$ and ending with $-100$ and the common difference is $-1$. So, the given expression can be written as
$\left( 100-1 \right)\left( 100-2 \right)\left( 100-3 \right)....\left( 100-199 \right)\left( 100-200 \right)=99\left( 98 \right)\left( 97 \right).....\left( -98 \right)\left( -99 \right)\left( -100 \right)$
Here we have a backward progression so there may be chances to get the value $0$. So, checking whether we have the zero in the given series or not. We can observe the value $100-100$ which comes in the middle of the series, and its value is equal to zero. So, substituting this value in the given expression, then we will get
$\left( 100-1 \right)\left( 100-2 \right)...\left( 100-100 \right)...\left( 100-199 \right)\left( 100-200 \right)=99\left( 98 \right)...0...\left( -99 \right)\left( -100 \right)$
We know that the product of any number with zero will be zero. Hence the value of the given expression becomes as
$\left( 100-1 \right)\left( 100-2 \right)\left( 100-3 \right)....\left( 100-199 \right)\left( 100-200 \right)=0$
So, the number of digits in the product is one.

So, the correct answer is “Option A”.

Note: In general students do not consider the term $100-100$ in the expression and calculate the product by doing a lot of calculations. It will result in time waste and loss of confidence. This kind of problem tests our intelligence.