Question

In how many ways can 14 different rings be worn on 5 fingers, such that any finger may have any number of rings?

Answer 1

Hint:: In this question, we have to find out the no. of ways of total no. of rings can be worn in $5$ fingers. In this question, no restrictions are given. Simply any rings can be worn in any fingers. So, all different rings can be worn in anyways in $5$ different fingers. As there are no restrictions regarding rings to be worn in which finger, so we can find the total no. of ways of wearing any ring in any finger by multiplying the total no. of fingers available for wearing rings (suppose $n$) for no. of times equal to no. of rings are there (suppose $r$) hence ${n^r}$will give the ultimate answer.

Complete step by step solution:
According to the question, $14$ different rings be worn on $5$ fingers.
Since any rings can be worn on any rings. There are no such restrictions that only one can be worn on one finger or etc.
According to the question:
One ring can be worn on any finger.
No. of fingers for $1st$ ring $ = 5$.
No. of fingers for $2nd$ ring $ = 5$.
No. of fingers for $3rd$ ring $ = 5$.
Similarly, No. of fingers for fourteenth $\left( {14th} \right)$ ring $ = 5$.

$\therefore$ No. of ways in which all $14$rings can be worn in $5$ fingers are $5 \times 5 \times 5 \times ......... \times 5$ (up to $14$)$ = {5^{14}}$ ways.

Additional information:
Note that the same formula would not be applied if there would be any restriction in the question regarding the wearing of rings in any of the selective fingers or there would be different types of rings; this point is mentioned here to make your concept crystal clear about where this can be or can’t be used.

Note: This is a question related to a conceptual topic i.e. permutation and combination. In this kind of problem focus especially on no. of places available (fingers here) & no. of things to be placed there is available (rings here) & if there are any restrictions regarding their no. of arrangements or ways given in the question or not. Do the cross-check of your answer once relating question with the formula you used to get the answer ( ${n^r}$here) attentively to make sure that you don’t make a mistake instead of knowing all the concepts to be applied & procedures to be used