A) \[\frac{1}{4}\]
B) \[\frac{1}{2}\]
C) \[\frac{1}{8}\]
D) \[\frac{1}{3}\]
Correct Answer: A
Solution :
Let \[l={{7}^{n}}+{{7}^{m}},\] then we observe that \[{{7}^{1}},{{7}^{2}},{{7}^{3}}\] and \[{{7}^{4}}\] ends in 7,9,3 and 1, respectively. Thus, \[{{7}^{i}}\] ends in 7,9,3 or 1 according as i is of the form 4k + 1, 4k + 2, 4k ? 1 or 4k, respectively. If S is the sample space, then \[n\left( S \right)={{\left( 100 \right)}^{2}}.\] \[{{7}^{m}}+{{7}^{n}}\] is divisible by 5, if (i) m is of the form 4k + 1 and n is of the form 4k ? 1 or (ii) m is of the from 4k + 2 and n is of the from 4k or (iii) m is of the form 4k ? 1 and n is of the from 4k + 1. (iv) m is of the form 4k and n is of the form 4k + 1. This, number of favourable ordered pairs \[\left( m,n \right)=4\times 25\times 25.\] Hence, required probability \[=\frac{4\times 25\times 25}{{{\left( 100 \right)}^{2}}}=\frac{1}{4}\]
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