Question

How many different words containing all the letters of
the word TRIANGLE can be formed so that
(i) consonants are never separated
(ii) consonants never come together
(iii) vowels occupy odd places.

Answer 1

$\large\underline{\sf{Solution-}}$

We know that,

If there are n distinct objects and out of these 'n' objects, 'r' objects have to be arranged in a line, then number of ways of such arrangement is

$\boxed{ \sf{ \:^nP_r = \frac{n!}{(n - r)!}}}$

and

$\boxed{ \sf{ \:^nP_n = {n!}}}$

Let's solve the problem now!!!

$\purple{\large\underline{\sf{Solution-(i)}}}$

We have given the word TRIANGLE.

Now, TRIANGLE word consist of 8 alphabets, including 3 vowels (A, E, I ) and 5 consonants ( T, R, N, G, L ).

So, taking 5 consonants ( T, R, N, G, L ) as one alphabet, we have now 4 letters, to be arranged. This can be arranged in P(4, 4) = 4! ways and corresponding to each of these arrangement, the consonants can be arranged together in P (5, 5) = 5! ways.

Hence,

Total number of words in which consonants are always together or never be separated is

$\rm \:  =  \:  \:4! \times 5!$

$\rm \:  =  \:  \:24 \times 120$

$\rm \:  =  \:  \:2880 \: ways$

$\green{\large\underline{\sf{Solution-(ii)}}}$

We have given the word TRIANGLE.

The total number of words that can be formed using all the 8 letters of the word ' TRIANGLE ' is P(8, 8) = 8 ! = 40320.

So, Total number of words in which consonants are never together

= Total number of words - Number of words in which consonants are always together

$\rm \:  =  \:  \:40320 - 2880$

$\rm \:  =  \:  \:37440 \: ways$

$\red{\large\underline{\sf{Solution-(iii)}}}$

We have given the word TRIANGLE.

Now, TRIANGLE word consist of 8 alphabets, including 3 vowels (A, E, I ) and 5 consonants ( T, R, N, G, L ).

In triangle word, there are 4 odd places and we wish to put 3 vowels in odd places.

So, These 3 vowels can be arranged themselves in P( 4, 3) = 4! ways and the remaining 5 consonants can arranged themselves in 5 places in P(5, 5) = 5! ways.

Hence,

Total number of ways in which vowels occupy odd places.

$\rm \:  =  \:  \:4! \times 5!$

$\rm \:  =  \:  \:24 \times 120$

$\rm \:  =  \:  \:2880 \: ways$