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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
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B
0.68
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C
0.068
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D
0.067
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Solution

The correct option is C $$0.067$$
The number of $$4$$ digit numbers is $$10000$$.
Let the number be $$abcd$$ , for it to be lucky number $$a+b=c+d$$ , $$0 \le a+b \le 18$$ , $$0 \le c+d \le18$$.
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 $$2 \times \cfrac{9 \times 10 \times 19}{6}+100=670$$.
The probability is $$\cfrac{670}{10000}=0.067$$.
(10≤n≤18)

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