Question

Find the number of different signals that can be generated by arranging at least $2$ flags in order (one below the other) on a vertical staff, if five different flags are available.

• A. $312$
• B. $313$
• C. $315$
• D. $320$

Answer 1

Answer: (D)

A signal can consist of either $2$ flags, $3$ flags, $4$ flags or $5$ flags. There will be as many $2$ flag signals as there are ways of filling in $2$ vacant places $\boxminus$ in succession by the $5$ flags available. By Multiplication rule, the number of ways is $5\times 4=20$.
in succession by Similarly, there will be as many $3$ flag signals as there are ways of filling in $3$ vacant places in succession by the $5$ flags. The number of ways is $5\times 4\times 3=60$. Continuing the same way, we find that The number of $4$ flag signals $=5\times 4\times 3\times 2=120$ and the number of $5$ flag signals $=5\times 4\times 3\times 2\times 1=120$ Therefore, the required number of signals $=20+60+120+120=320$.