Question

A four digit number is said to be lucky if sum of its first two digits is equal to the sum of its last digits. If a four digit number is picked up at random, find the probability that it is lucky number.

• A. $0.67$
• B. $0.68$
• C. $0.068$
• D. $0.067$

Answer 1

Answer: (D)

$0.067$

The number of $4$ digit numbers is $10000$.

Let the number be $abcd$ , for it to be lucky number $a+b=c+d$ , $0\le a+b\le 18$ , $0\le c+d\le 18$.

The number of non negative solutions of $x+y=n$ with $0\le x,y\le 9$ is $n+1$ and for $10\le x,y\le 18$ is $19-n$.

The number of ways to choose the numbers are $1\times 1+2\times 2+3\times 3+......+10\times 10+9\times 9+......+1\times 1$ .

By using sum of squares of natural numbers , we get $2\times \frac{9\times 10\times 19}{6}+100=670$.

The probability is $\frac{670}{10000}=0.067$.

(10≤n≤18)