The probability that in a random arrangement of the letters of the word $UNIVERSITY$ the $2$ Is come together is
- $\displaystyle \frac{3}{5}$
- $\displaystyle \frac{2}{5}$
- $\displaystyle \frac{1}{5}$
- $\displaystyle \frac{4}{5}$
The word UNIVERSITY has 10 letters with two I's. Treating the two I's as a single unit, we arrange 9 units in 9!/2! ways (since there are two I's, but we treat them as one block, wait: the letters are U, N, I, V, E, R, S, I, T, Y. There are 10 letters, two I's. Total arrangements = 10!/2!. Arrangements with I's together = 9! ways. Probability = 9! / (10!/2!) = 2/10 = 1/5.
The word UNIVERSITY contains 10 letters in total, with the letter I appearing exactly twice. The total number of possible arrangements for all the letters is 10! divided by 2! to account for the repeated I, which equals 1814400. To find the arrangements where the two I's come together, we treat the two I's as a single combined block or letter. This reduces the total number of items to arrange to 9, and these 9 distinct items can be arranged in 9! ways, which equals 362880. The probability is the ratio of favorable arrangements to total arrangements, calculated as 9! divided by (10! / 2!), which simplifies to (9! multiplied by 2!) divided by (10 multiplied by 9!), resulting in 2/10 or 1/5.