Question

The number of ways in which 20 differently coloured flowers be strung in the from of a garland is:-
A.191 B. $\dfrac{{19!}}{2}$ C. 201 D. 211

Answer 1

Hint: Try to remember that the number of ways of arranging n distinct objects in a circle is \[{\rm{(n - 1)!}}\]. But for a garland the answer is halved. Apply this concept to reach the result.

Complete step-by-step answer:
As mentioned above the numbers of ways will be halved. Therefore total number of ways is,
$\frac{{{\rm{(n - 1)!}}}}{{\rm{2}}}$
Now let’s analyse the values given to us in the question.
Given there are 20 distinct coloured flowers the number of ways in which we can form the garland is given by,
$\frac{{(20 - 1)!}}{2}$
We are subtracting 1 from 20 because we will fix a flower first and then only arrange the other flowers around it.
On solving we get,
\[ = \,\frac{{19!}}{2}\]

Hence, the correct option is B.

Note:- It should be noted that though we have (n-1)! Combinations for a circular arrangements but for objects such as ring, necklace, garland the combinations go down to $\dfrac{{{\text{(n - 1)!}}}}{{\text{2}}}$. This is because either in clockwise or anticlockwise direction these objects look identical hence the total combinations are halved.