Question

Number of ways of selecting $2$ numbers form integers $1$ to $100$ if their sum is divisible by $5$ are

• A. 980
• B. 990
• C. 1000
• D. None of these

Answer 1

Answer: (B)

The numbers can be categorized as
(a) $5,10,15,\dots ,100\to 5\lambda$
(b) $1,6,11,\dots ,96\to 5\lambda +1$
(c) $2,7,12,\dots ,97\to 5\lambda +2$
(d) $3,8,13,\dots ,98\to 5\lambda +3$
(e) $4,9,14,\dots ,99\to 5\lambda +4$
Now, sum of two numbers is divisible by 5 then we can have following cases
(i) both the numbers from the list (a) - number of selection ways are ${}^{20}{C}_2$
(ii) one number from list (b) and other from list (e) - number of selection ways are ${}^{20}{C}_1\times {}^{20}{C}_1$
(iii) one number from list (c) and other from list (d) - number of selection ways are ${}^{20}{C}_1\times {}^{20}{C}_1$
Hence, total number of ways are
${}^{20}{C}_2+{}^{20}{C}_1\times {}^{20}{C}_1+{}^{20}{C}_1\times {}^{20}{C}_1$
$=990$