Question

The number of ways in which the letters of the word TRIANGLE can be arranged such that two vowels not occur together is

• A. $1200$
• B. $2400$
• C. $14400$
• D. none of these

Answer 1

The correct option is C $14400$

The word ${}^{\prime \prime }TRIANGL{E}^{\prime \prime }$ has $8$ letters of which $3$ are vowels.

$\therefore \text{Total words}=8!=8\times 7\times 6\times 5\times 4\times 3\times 2\times 1=40320$

No. of words in which $two$ vowels are together. We select $two$ vowels and then tie them together so that we can effectively left with $7$ letters and also we need to take care of internal arrangement of two vowels.

$\therefore \text{No. of words in which two vowels are together}=3C_2\times 7!\times 2!=3\times 7\times 6\times 5\times 4\times 3\times 2\times 1\times 2=30240$

But, we need to include words in which $three$ vowels are together.

$\therefore \text{No. of words in which three vowels are together}=3C_3\times 6!\times 3!=4320$

$\therefore \text{Required no. of words}=40320-30240+4320=14400$