[tex]\large\underline{\sf{Solution-}}[/tex]
We know first 11 whole numbers are 0, 1, 2, 3, 4, ..., 10
Now, we have to chose 2 numbers x and y from these 11 numbers.
So, Number of ways to chose x and y = 11 × 11 = 121
Now, further given that we have to find x and y such that
[tex]\rm \: |x - y| \leqslant 5 \\ [/tex]
Now, Case :- 1 When x = 0
[tex]\rm \: |0 - y| \leqslant 5 \\ [/tex]
[tex]\rm \: | - y| \leqslant 5 \\ [/tex]
[tex]\rm \: |y| \leqslant 5 \\ [/tex]
[tex]\bf\implies \: - 5 \leqslant y \leqslant 5 \\ [/tex]
[tex]\rm\implies \:y \: = \{0,1,2,3,4,5 \} \\ [/tex]
Case :- 2 When x = 1
[tex]\rm \: |1 - y| \leqslant 5\rm\implies \: - 5 \leqslant 1 - y \leqslant 5 \\ [/tex]
[tex]\rm\implies \: - 6 \leqslant - y \leqslant 4[/tex]
[tex]\rm\implies \: - 4\leqslant y \leqslant 6 \\ [/tex]
[tex]\rm\implies \:y \: = \{0,1,2,3,4,5,6 \} \\ [/tex]
Case :- 3 When x = 2
[tex]\rm \: |2 - y| \leqslant 5\rm\implies \: - 5 \leqslant 2 - y \leqslant 5 \\ [/tex]
[tex]\rm\implies \: - 7 \leqslant - y \leqslant 3[/tex]
[tex]\rm\implies \: - 3 \leqslant y \leqslant 7 \\ [/tex]
[tex]\rm\implies \:y \: = \{0,1,2,3,4,5,6,7 \} \\ [/tex]
Case :- 4 When x = 3
[tex]\rm \: |3 - y| \leqslant 5\rm\implies \: - 5 \leqslant 3 - y \leqslant 5 \\ [/tex]
[tex]\rm\implies \: - 8\leqslant - y \leqslant 2 \\ [/tex]
[tex]\rm\implies \: - 2\leqslant y \leqslant 8 \\ [/tex]
[tex]\rm\implies \:y \: = \{0,1,2,3,4,5,6,7,8 \} \\ [/tex]
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Similarly on proceeding like this, x = 4, 5, 6, 7, 8, 9, 10, the corresponding values of y are 9, 10, 11, 10, 9, 8, 7, 6.
Total number of pairs of x and y = 6 + 7 + 8 + 9 + 10 + 11 + 10 + 9 + 8 + 7 + 6 = 91
So, required probability is given by
[tex]\rm\implies \:P( |x - y| \leqslant 5) = \frac{91}{121} \\ [/tex]