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Q.

Messages are conveyed by arranging four white, one blue, and three red flags on a pole. Flags of the same color are alike. If a message is transmitted by the order in which the colours are arranged, the total number of messages that can be transmitted if exactly six flags are used is

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a

125

b

185

c

65

d

45

answer is D.

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Detailed Solution

We will consider the following cases:

Case	Flags	No. of signals
4 alike and 2 others alike	4 white and 2 red	$\frac{6!}{4! 2!} = 15$
4 alike and 2 others different	4 white, 1 red and 1 blue	$\frac{6!}{4!} = 30$
3 alike and 3 others alike	3 white, 3 red	$\frac{6!}{3! 3!} = 20$
3 alike and 2 other alike and 1 different	3 white, 1 blue, 2 red or 3 red, 1 blue,2 white	$^{2} C_{1} \times \frac{6!}{3! 2!} = 120$
	Total	185

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