Question

Four numbers are chosen at random from $\{1,2,3,......,40\}$. The probability that they are not consecutive is

• A. $\frac{1}{2470}$
• B. $\frac{4}{7969}$
• C. $\frac{2469}{2470}$
• D. $\frac{7965}{7969}$

Answer 1

The correct option is C $\frac{2469}{2470}$

Probability when all four chosen numbers are not consecutive $=$ 1 $-$ Probability when all four chosen numbers are consecutive
Number of cases when all selected numbers are consecutive $=37$ $\left[\right(1,2,3,4);(2,3,4,5);........(37,38,39,40\left)\right]$
Total number of cases of selecting four numbers out of 40 ${=}^{40}{C}_4$

Probability when all four chosen numbers are consecutive $=\frac{37}{{}^{40}{C}_4}=\frac{1}{2470}$

Probability when all four chosen numbers are not consecutive $=1-\frac{1}{2470}=\frac{2469}{2470}$