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# I have $4$ pants, $3$ shirts and $2$ banians, In how many ways can I put them on?

Last updated date: 29th Feb 2024
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Hint: Based on this problem we have a combination formula in which we arrange (or) by selecting the objects appropriately.
Here we will use the combination formula in this problem.
Formula: $n!\, = \,n\, \times \,\left( {n\, - \,1} \right)\, \times \,\left( {n\, - \,2} \right)\, \times \,...\, \times \,3\, \times \,2\, \times \,1$
$^n{C_{r\,}} = \,\dfrac{{n!}}{{\left( {n - r} \right)!r!}}\,\,\,;\,\,0 \leqslant r < n$ And
$^n\,{C_{o\,}} = n{C_n}\, = 1$
Also, $\,^n\,{C_1}\, = \,n$

Given that: $4\,$ pants, $3$ shirts and $2$ banians
$\left( {^4\,{C_1} \times\,^3{C_1}\, \times \,^2{C_1}} \right)$ ways
Now, expanding the above expression using the formula $^n{C_1}\, = \,n$ , we get $^4{C_1} = 4,\,\,^3{C_1} = 3\,$ and $^2{C_1}\, = 2$
$\left( {4\, \times \,3\, \times \,2} \right)$ ways
So, therefore we can conclude that the number of ways which he can put them on is $\,24$ ways.
Note: The number of clothes in which he can put on him is $\,24$ ways, Let us discuss the above problem as recap (or) review it at glance. Here in this problem, if he has $4$ pants, then he has to select all the four pants which he has to wear on him. That is, he can select the $4$ pants in $4$ ways, similarly, if he has $3$ shirts, then he instantly pick/select the $3$ shirts in $3$ ways, also if he has $2$ baniyans and has to select it then he can do it in $2$ ways. Thus, in general if he picks $a,b,$ and $c$ objects all one at a time then, the total number of ways will be chosen as $\left( {a\, \times \,b\, \times \,c} \right)$ ways. And similarly here we arrive at the conclusion as the total number of ways he can put the cloth on him is $\left( {4\, \times \,3\, \times \,2} \right)$ ways which is equals to $24$ ways.