Question

The letters of the word ‘UNIVERSITY’ are written in all possible ways taking care that two I’s do not occur together. The possible number of ways is
A.${\text{10!}}$
B.$\dfrac{{\text{1}}}{{\text{2}}}{\text{10!}}$
C.${\text{9!}}$
D.${{4 \times 9!}}$

Answer 1

Hint: We can obtain the number of ways the letters can be written such that 2 I’s do not occur together by calculating the number of ways the letter can be rearranged with the 2 I’s together and subtracting it from the total number of ways. The total number of ways can be found using the formula $\dfrac{{{\text{n!}}}}{{{\text{r!}}}}$ and the number of ways of arranging letters with 2 I’s together can be found by taking the 2 I’s as one letter.

Complete step-by-step answer:
First, we need to find the total number of ways the letters can be written. It is given by
$\dfrac{{{\text{n!}}}}{{{\text{r!}}}}$ where n= number of letters and r = number of repeating letters
So, the total number of ways the letters of word ’UNIVERSITY’ are written ${\text{ = }}\dfrac{{{\text{10!}}}}{{{\text{2!}}}}$
Now, we need to find the number of ways the letters can be written with the 2 I’s together. Let us consider the 2 I’s together. The other 8 letters and the 2 I ‘s can be arranged in 9! ways.
We need to find the number of ways the letters can be written such that 2 I’s do not occur together. For that we subtract the number of ways of writing 2 I’s together from the total number of ways. Thus, we get, ${\text{P = }}\dfrac{{{\text{10!}}}}{{{\text{2!}}}}{\text{ - 9!}}$
On further simplification, we get,
${\text{P = 9!}}\left( {\dfrac{{{\text{10}}}}{{\text{2}}}{\text{ - 1}}} \right){\text{ = 9!}}\left( {\dfrac{{{\text{10 - 2}}}}{{\text{2}}}} \right){{ = 4 \times 9!}}$
Hence there are 4 X 9! ways in which the letters can be written such that 2 I’s don’t occur together.
Therefore, the correct answer is option D.

Note: For finding the number of ways of arranging letters, we must consider the repeating letters and it must be used in the equation accordingly. It is a common error to skip the repeating letters. Drawing the places to be filled and visualizing can help for better understanding of the solution. The concept used here is permutations and combinations.