Question

The number of ways in which four numbers can be selected from $1,2,3,\dots ,30$ so that they are not consecutive is

• A. ${}^{30}{C}_4-26$
• B. ${}^{30}{C}_4-27$
• C. ${}^{30}{C}_4-28$
• D. ${}^{30}{C}_4-29$

Answer 1

The correct option is B ${}^{30}{C}_4-27$

The number of ways of selecting four numbers from $1$ to $30$ without any restriction is ${}^{30}{C}_4$.

The number of ways of selecting four consecutive number
$[\text{i.e.},(1,2,3,4),(2,3,4,5),...,(27,28,29,30\left)\right]$is $27$.

Hence, the number of ways of selecting four integers which excludes consecutive four selections
$={}^{30}{C}_4-27$