pls help simple question (1 Viewer)

[sharoooooo]

In how many ways can four consonants and three vowels be arranged in a row:
a) so that the three vowels are always together
b) so that the first and last places are occupied by consonants?

pls help me, I've just started doing perms and combs and im lost
The answer for a) 720, b) 1440

my working for a) 3! x 4! x 4 = 576
why doesnt this work?? i dont understand why 576 is not the answer
help me pls
thank you

 

[InteGrand]

In how many ways can four consonants and three vowels be arranged in a row:
a) so that the three vowels are always together
b) so that the first and last places are occupied by consonants?

pls help me, I've just started doing perms and combs and im lost
The answer for a) 720, b) 1440

my working for a) 3! x 4! x 4 = 576
why doesnt this work?? i dont understand why 576 is not the answer
help me pls
thank you

For a) it should be ×5 at the end rather than ×4, since the 'group' of 3 vowels can be in any of 5 places. i.e.:

[v₁, v₂, v₃],c₁,c₂,c₃,c₄

Treat that bracketed group of vowels as one object, then there's 5 'objects', so 5! ways to arrange them, and for each arrangement, we can arrange the vowels in the brackets in 3! ways, so the answer is 5! × 3! = 720.

For b), We can arrange the endpoints with consonants in ⁴P₂ = 12 ways. Then, just arrange the remaining 5 letters in the middle, which is done in 5! ways. Hence the answer is 5!×12 = 1440.

 

[sharoooooo]

ahh i see now
thank you so much!!