  Question

A four digit number (numbered from 0000 to 9999) is said to be lucky if 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 find the probability that it is lucky number.

A
Required probability=0.067  B
Required probability=0.077  C
Required probability=0.167  D
Required probability=0.177  Solution

The correct option is A Required probability$$=0.067$$A 4 digit number can be written as $$1000(a)+100(b)+10(c)+d$$It is lucky if $$(a+b)=c+d=n$$ ...(i)If n=0, total combination is 1.If n=1 total combinations are 4If n=2 total combinations are 9::Again if n=9 total combinations are 100If n=10  total combinations are 81and so on till the total combination is again 1.Hence Total number of lucky numbers is $$=1^{2}+2^{2}+...10^{2}+9^{2}+8^{2}+...1^{2}$$$$=2(1^{2}+2^{2}+....10^{2})-10^{2}$$$$=670$$Total numbers will be $$10000$$Hence the required probability is $$=\dfrac{670}{10000}$$$$=0.067$$Maths

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