Question

There are $8$ events that can be scheduled in a week. Then total number of ways that these $8$ events are scheduled on exactly $6$ days of a week is given by $266\times k!,$ where $k\in N.$ The value of $k$ is

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is C $7$

There are $8$ events to be scheduled in $6$ days.
This can be done in two ways.

Case I:
$3$ events are scheduled on one of the day and rest each day has $1$ event each.
$3$ events out of $8$ events can be selected in number of ways $={}^8{C}_3$
Now, these events can be arranged in $6$ days out of $7$ days of week in number of ways $={}^7{P}_6$
Number of ways $=\frac{8!}{5!\times 3!}\times 7!=56\times 7!$

Case II:
$2$ events are scheduled on couple of days and rest each day has $1$ event each.
Pair of $2$ events out of $8$ events can be selected in number of ways $=\frac{8!}{4!\times 2!\times 2!\times 2!}$
Now, these events can be arranged in $6$ days out of $7$ days of week in number of ways $={}^7{P}_6$
Number of ways $=\frac{8!}{4!\times 2!\times 2!\times 2!}\times 7!=210\times 7!$

Therefore, total number of ways $=266\times 7!$
$\therefore k=7$