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 $= 20 + 60 + 120 + 120 = 320$.