Question

How many words can be formed with the letters of the word 'UNIVERSITY', the vowels remaining together?

Answer 1

In the word 'UNIVERSITY' there are 10 letters of which 2 are 1's.
There are 4 vowels in the given word of which 2 are 1's.
These vowels can be put together in $\frac{4!}{2!}$
ways.
Considering these 4 vowels as one letter there are 7 letters which can be arranged in 7! ways.
Hence, by fundamental principle of multiplication, the required number of arrangements is
= $\frac{4!}{2!}\times 7!$
= $\frac{4\times 3\times 2}{2!}\times 7\times 6\times 5\times 4\times 3\times 2$
= $4\times 3\times 7\times 6\times 5\times 4\times 3\times 2$
= 60480.