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).

Answer 1

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.