A four digit number (numbered from 0000 to 9999) is said to be lucky if the sum of its 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 number is

A
$0.065$
B
$0.064$
C
$0.066$
D
$0.067$

Solution:

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 $0 \leq x_{1} + x_{2} \leq 18$ and $0 \leq x_{3} + x_{4} \leq 18$ .

Also, the number of non-negative integral solutions of $x + y = m$

(with $0 \leq x, y \leq 9$ ) is $\{\begin{cases} m + 1 i f 0 ⩽ m \leq 9 \\ 19 - m i f 10 \leq m \leq 18 \end{cases}$

$\begin{array}{l} The number of favourable ways = 1 \times 1 + 2 \times 2 + \dots + 10 \times 10 + 9 \times 9 + 8 \times 8 + \dots \\ + 1 \times 1 \\ = 2 \{\frac{9 \times 10 \times 19}{6}\} + 100 = 670 \end{array}$ n(E) = 670

$∴$ probability of required event $= \frac{670}{1000} = 0.067 .$

A four digit number (numbered from $0000$ to $9999$ ) is said to be lucky if the sum of its 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 number is

Solution:

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