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The question is- Out of 10 different flowers of different colours, how many different garlands can be formed if each garland consists of 6 flowers of different colours?

Now, they said that this can be done in ${10 \choose 6}$ $\frac {5!}{2}$ way. This is because clockwise and anticlockwise arrangements are the same. Now, my question is why should we arrange them in the first place if their arrangements in clockwise and anticlockwise manner are the same.

Can't we simply write $10 \choose 6$? Because, the number of ways I can select 6 colours is the number of way I can make a garland.

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    A garland is a circular ordering of flowers. Presumably, they want to count rotations and reflections of the order as the same. There are a of different ways to order the flowers after you've picked the six colors, so no, $\binom{10}{6}$ is not good enough. – 2017-02-22
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    I have got the logic now. Thanks for the help though. – 2017-02-22

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There are $\binom{10}{6}$ ways to select $6$ colours from $10$ and $6!$ ways to order those $6$, but only half of those are counted because the clockwise and anticlockwise patterns are the same.

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If the order of the flowers in the garland didn't matter, then the answer would be $10 \choose 6$. But since rose-carnation-daisy is different than daisy-rose-carnation, then order does matter. How many ways to order 6 flowers? $5!$. But you said that clockwise and counter-clockwise arrangements are the same, so we have to divide by 2.

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    Since counterclockwise and clockwise arrangements are the same, you should probably explain more carefully why you initially treat rose-carnation-daisy as different from daisy-carnation-rose. – 2017-02-22
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    @ErickWong Thanks for the suggestion. I just edited the order of the flowers. I couldn't think of 6 different kinds of flowers! – 2017-02-22
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    Thanks a ton for the help. I really appreciate it. – 2017-02-22
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A garland with red, pink, white, yellow, purple, blue in clockwise order has the colors red, blue, purple, yellow, white, pink in anticlockwise order. Or you could say it has colors in order purple, yellow, white, pink, red, blue if you start with the purple flower instead of the red one.

But no matter which flower you start with, and no matter whether you go around clockwise or anticlockwise, the colors will never come out red, white, pink, yellow, purple, blue. The pink flower will always be next to the red flower, and the white one never will, when you examine this particular garland.

The number $\frac{5!}{2}$ is the number of different arrangements of the chosen six colors, such that no two of those arrangements will list the colors in the same order as any other, no matter which flower you start with and whether you go around clockwise or anticlockwise.