Probability 1 Question 5
5. If two different numbers are taken from the set ${0,1$, $2,3, \ldots, 10}$, then the probability that their sum as well as absolute difference are both multiple of 4 , is
(a) $\frac{6}{55}$
(b) $\frac{12}{55}$
(c) $\frac{14}{45}$
(d) $\frac{7}{55}$
(2017 Main)
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Answer:
Correct Answer: 5. (a)
Solution:
- Total number of ways of selecting 2 different numbers from ${0,1,2, \ldots, 10}={ }^{11} C _2=55$
Let two numbers selected be $x$ and $y$.
$$ \begin{array}{lc} \text { Then, } & x+y=4 m \\ \text { and } & x-y=4 n \\ \Rightarrow & 2 x=4(m+n) \text { and } 2 y=4(m-n) \\ \Rightarrow & x=2(m+n) \text { and } y=2(m-n) \end{array} $$
$\therefore x$ and $y$ both are even numbers.
| $x$ | $y$ |
|---|---|
| 0 | 4,8 |
| 2 | 6,10 |
| 4 | 0,8 |
| 6 | 2,10 |
| 8 | 0,4 |
| 10 | 2,6 |
$\therefore$ Required probability $=\frac{6}{55}$
