Question

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 a lucky number is

• A. $0.07$
• B. $0.08$
• C. $0.67$
• D. $0.067$

Answer 1

Answer: (D)

Let the number be $x_1,x_2,x_3,x_4$
$x_1+x_2+x_3+...x_r=n$
The number of non-negative integral solution including zero will be
${}^{n+r-1}{C}_{r-1}$
For $x_1+x_2=0$ the no. of solution will be $1$
$x_1+x_2=1$ no. of solution will be $2$
$x_1+x_2=2$ no. of solution will be $3$
Hence for $x_1+x_2=n$ no. of solution will be $n+1$
Similarly for $x_3+x_4=n$ no. of solution will be $n+1$
Hence no.of 4 digit no will be $(n+1{)}^2$
Now for $x_1+x_2=10,11,12,13,\mathrm{14...18}$ we have to consider
$0\le x_1\le 9$ and $0\le x_2\le 9$
For $10$, the answer will be the coefficient of $x^{10}$ in the expansion of
$(1+x+x^2...x^9{)}^2$
$=11-2=9$
Similarly for $11$ it will be $8$ and so on.
Hence corresponding to the number of lucky numbers we get
$=(1^2+2^2+3^2..{.10}^2)+(1^2+2^2+3^2..{.9}^2)$
$=2(1^2+2^2+3^2..{.9}^2)+{10}^2$
$=570+100$
$=670$
The total number of integers between $0000$ and $9999$ are
$10000$.
Hence Probability will be $\frac{670}{10000}$
$=0.067$