I have a question that says: Find the number of different words that can be formed from the letters of the
word ‘TRIANGLE’ so that no vowels are together
I did this in the following way:
Number of ways in which TRIANGLE can be arranged –
Number of ways in which TRIANGLE can be arranged where 3 vowels are together –
Number of ways in which TRIANGLE can be arranged where 2 vowels are together = 8! – (6!*3!) – (7!*2!*3) = 5760
The correct answer in my book is given to be 14400 calculated as:
xTxRxNxGxLx : x depicts spaces where vowels can be arranged and they are not together.
Therefore, Number of ways in which Consonants can be arranged*Number of ways in which vowels can be arranged = 5!6C33! = 14400.
Can anyone help me figure out why the method I follow isn't working?
Best Answer
Applying inclusive - exclusive principle
Making group with two vowels$(7!\times 2!\times 3)$ will cover also making group with three vowels $(6!\times3!)$
for example $TR(IA)ENGL$ arrangements of $(IA)$ together also has $E(IA)$ and $(IA)E$
$$8! - (7!\times 2!\times 3) + (6!\times3!)= 14400$$