Question

In how many ways 7 pictures can be hung from 5 picture nails on a wall?

Answer 1

In this question we have to find the number of ways to hang 7 pictures from 5 picture nails on a wall, it means we have to choose 5 nails at a time. To solve this question we will use the concept of permutation. The basic permutation formula is given as
${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$
Where, n is the total number of objects and r is the number of choices.

Complete step by step solution:
We have to find the number of ways to hang 7 pictures from 5 picture nails on a wall.
Now, we know that a permutation is the arrangement of a set of data in some specific order. If we have total number of n datasets and we have to choose r objects from the dataset then the basic permutation formula is given as:
${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$
Where, n is the total number of objects and r is the number of choices.
Here we have a total 7 numbers of objects and we have to choose 5 at a time.
So the total number of ways to choose 5 out of 7 pictures will be
$\Rightarrow {}^{7}{{P}_{5}}=\dfrac{7!}{\left( 7-5 \right)!}$
Now, simplifying the above obtained equation we will get
$\begin{aligned} & \Rightarrow {}^{7}{{P}_{5}}=\dfrac{7\times 6\times 5\times 4\times 3\times 2!}{2!} \\\ & \Rightarrow {}^{7}{{P}_{5}}=7\times 6\times 5\times 4\times 3 \\\ & \Rightarrow {}^{7}{{P}_{5}}=2520 \\\ \end{aligned}$
Hence we get 2520 ways to hang 7 pictures from 5 picture nails on a wall.

Note: Here in this question we can also use the concept of combination. First we will find the number of ways of selecting the 5 pictures out of and then find the number of ways to arrange 5 pictures on the 5 picture nails. Then multiplying both will give the desired answer.
Now, number of ways to select 5 pictures out of 7 will be

& \Rightarrow {}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!} \\\ & \Rightarrow {}^{7}{{C}_{5}}=\dfrac{7!}{5!\left( 7-5 \right)!} \\\ \end{aligned}$$ Now, simplifying the above obtained equation we will get $\begin{aligned} & \Rightarrow {}^{7}{{C}_{5}}=\dfrac{7\times 6\times 5!}{5!\times 2!} \\\ & \Rightarrow {}^{7}{{C}_{5}}=\dfrac{7\times 6}{2\times 1} \\\ & \Rightarrow {}^{7}{{C}_{5}}=21 \\\ \end{aligned}$ So there are a total 21 ways of selecting 5 pictures. Now, total number of ways in which 5 pictures can be arranged on 5 picture nails will be $5!$ Now, the number of ways to hang 7 pictures from 5 picture nails on a wall will be $\begin{aligned} & \Rightarrow 21\times 5\times 4\times 3\times 2\times 1 \\\ & \Rightarrow 2520 \\\ \end{aligned}$ Hence above is the required answer.