Question

# Find the number of ways in which 5 beads, chosen from 8 different beads can be threaded on a ring.

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Hint: At first count the number of ways we can choose 5 beads from 8 different beads. Then count in how many ways we can arrange them.

It is given in the question that we have 8 different beads.
We know that the combination is a way of selecting items from a collection, such that the order of selection does not matter.
At first when we are selecting 5 beads, from 8 different beads then the order of selection does not matter.
So, we can choose 5 beads from 8 different beads in ${}^{8}{{C}_{5}}$ ways.
${}^{8}{{C}_{5}}=\dfrac{8!}{\left( 8-5 \right)!\times 5!}=\dfrac{8!}{3!\times 5!}=\dfrac{5!\times 6\times 7\times 8}{1\times 2\times 3\times 5!}=\dfrac{6\times 7\times 8}{2\times 3}=56$
Therefore, we can select 5 beads from 8 different beads in 56 ways.
Now we have to arrange those 5 beads on a circular ring.
When we will arrange these 5 beads then the order of arranging them plays a very significant role. If the order changes, that will become a different arrangement. Therefore, we have to use permutation.
Permutation is arranging all the members of a set into some sequence or order, where the order of selection matters.
Now, we can arrange $n$ elements in a circular permutation in $\left( n-1 \right)!$ ways.
So we can arrange 5 beads in a circular ring in $\left( 5-1 \right)=4!=1\times 2\times 3\times 4=24$ ways.
But here we can arrange 5 beads in two directions. One is clockwise, another is anticlockwise. Here in both directions we will get the same arrangement. So, we have to divide 24 by 2.
Therefore the total number of different ways of arranging 5 beads is $\dfrac{24}{2}=12$ .
Now, we can select 5 beads from 8 different beads in 56 ways.
In each way we can arrange them in 12 ways.
So, the total number of ways of selecting 5 beads from 8 different beads and arranging them on a circular ring is $56\times 12=672$
Therefore the total number of ways in which 5 beads, chosen from 8 different beads be threaded on a circular ring is 672.

Note: For circular permutation we have to be careful about the clockwise and anticlockwise directions.
Generally we make a mistake here. If the direction does not matter then we have to divide $\left( n-1 \right)!$ by 2.