Permutation Combination Question 13
Question 13 - 30 January - Shift 2
The number of ways of selecting two numbers $a$ and $b$, $a \in{2,4,6, \ldots ., 100} \quad$ and $b \in{1,3,5, \ldots ., 99}$ such that 2 is the remainder when $a+b$ is divided by 23 is
(1) 186
(2) 54
(3) 108
(4) 268
Show Answer
Answer: (3)
Solution:
Formula: Combination under restriction
$a \in{2,4,6,8,10, \ldots, 100}$
$b \in{1,3,5,7,9, \ldots \ldots, 99}$
Now, $a+b \in{25,71,117,163}$
(i) $a+b=25$, no. of ordered pairs $(a, b)$ is 12
(ii) $a+b=71$, no. of ordered pairs $(a, b)$ is 35
(iii) $a+b=117$, no. of ordered pairs $(a, b)$ is 42
(iv) $a+b=163$, no. of ordered pairs $(a, b)$ is 19
$\therefore$ total $=108$ pairs
