Question

A four digit number (numbered from 0000 to 9999) is said to be lucky if the sum of first two digits is equal to the sum of its last two digits. If a four digit number is picked up at random, then the probability that it is lucky is

• A. (a) >0.065
  
• B. (b) <0.068
  
• C. (c) 0.066
  
• D. (d) 0.067

Answer 1

Answer: (A), (B), (D)

The correct options are
A

(a) >0.065

B

(b) <0.068

D

(d) 0.067

The total number of ways of choosing the ticket is 10000
Let the four digits number on the ticket be $x_1{x}_2{x}_3{x}_4.$
Note that $0\le x_1+x_2\le 18and0\le x_3+x_4\le 18.$
Also, the number of non-negative integral solutions of x+y=m with $0\le x,y\le 9$ is $m+1$ if $0\le m\le 9$ and
is $19-m$ if $10\le m\le 18$
Thus, the number of favourable ways
$=1\times 1+2\times 2+...........+10\times 10+9\times 9+8\times 8+........+1\times 1=2\left\{\frac{9\times 10\times 19}{6}\right\}+100=670$
$\therefore$ Probability of required event= $\frac{670}{10000}=0.067$
Note that 0.067 > 0.065 and 0.067 < 0.068.