How many different signals can be transmitted by arranging \[2\] red, \[3\] yellow and \[2\] green flags on a pole? (Assume that all the \[7\] flags are used to transmit a signal).

Question

How many different signals can be transmitted by arranging $$2$$ red, $$3$$ yellow and $$2$$ green flags on a pole? (Assume that all the $$7$$ flags are used to transmit a signal).

Vedantu Content Team · Accepted Answer

Hint:In order to determine the different ways signals can be transmitted.The signals can be transmitted by arranging $$2$$red, $$3$$ yellow and $$2$$ green flags on a pole. We use the factorial method to solve the problem. The product of a whole number $$'n'$$ with every whole number until $$1$$ is called the factorial. Finally, we obtain the required solution. Complete step by step answer: In this given problem, the statement includes the given as $$2$$ red, $$3$$ yellow and $$2$$ green flags to be in a pole. Here, We use factorial method for solving the given problem,The total number of ways to arrange if all are different = $$7!$$But here $$3$$ yellows are the same and $$2$$ reds and $$2$$ green are also the same.Thus, the number of ways is $$ = \dfrac{{7!}}{{3! \times 2! \times 2!}}$$Expanding the factorial terms to find the value of it.$$\dfrac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{(3 \times 2 \times 1) \times (2 \times 1) \times (2 \times 1)}}$$By simplify the multiplication of  the fraction, then$$\dfrac{{5040}}{{(6) \times (2) \times (2)}}$$Multiplying on the values to find the value of the number of ways,$$\dfrac{{5040}}{{24}}$$The number of ways is $$ = 210$$Thus, the number of ways to arrange the flags in the pole is $$210$$. The solution for the above question in which they asked for different signals to be transmitted by arranging $$2$$ red, $$3$$ yellow, and $$2$$ green flags on the pole.Based on the calculation using factorial ways the answer to be found as it can be arranged in $$210$$ ways. Therefore,  $$210$$ different signals can be transmitted by arranging $$2$$ red, $$3$$ yellow and $$2$$ green flags on a pole.Note: Some of the interesting facts about Factorials they are:-The number of zeros at the end of n! is roughly $$\dfrac{n}{4}$$-$$70!$$ is the smallest factorial larger than a googol.-The sum of the reciprocals of all factorials is $$e$$. -Factorials can be extended to fractions, negative numbers and complex numbers by the gamma function.