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How many number of different signals can be generated by arranging atleast 2 flags in order (one below the other) on vertical staff, if 4 different flags are available?

How many number of different signals can be generated by arranging atleast 2 flags in order (one below the other) on vertical staff, if 4 different flags are available? Correct Answer 60

Concept:

Fundamental principal of multiplication:

Let us suppose there are two tasks A and B such that the task A can be done in m different ways following which the second task B can be done in n different ways. Then the number of ways to complete the task A and B in succession respectively is given by: m × n ways

Fundamental principal of addition :

Let us suppose there are two tasks A and B such that the task A can be done in m different ways and task B can be completed in n ways. Then the number of ways to complete either of the two tasks is given by: (m + n) ways.

Calculation:

Here, a signal can have either 2 flags, 3 flags or 4 flags.

Case – 1: When the signal has 2 flags one below the other on a vertical staff.

The number ways to fill the 1st vacant place = 4

The number ways to fill the 2nd vacant place = 3

∴ No. of ways to generate a signal with 2 flags = 4 × 3 = 12.

Case – 2: When the signal has 3 flags one below the other on a vertical staff.

The number ways to fill the 1st vacant place = 4

The number ways to fill the 2nd vacant place = 3

The number ways to fill the 3rd vacant place = 2

∴ No. of ways to generate a signal with 3 flags = 4 × 3 × 2 = 24

Case – 3: When the signal has 4 flags one below the other on a vertical staff.

The number ways to fill the 1st vacant place = 4

The number ways to fill the 2nd vacant place = 3

The number ways to fill the 3rd vacant place = 2

The number ways to fill the 4th vacant place = 1

∴ No. of ways to generate a signal with 4 flags = 4 × 3 × 2 × 1 = 24

Hence, total no. of signals that can be generated = 12 + 24 + 24 = 60

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