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How many vowels are there ?

all vowels are there

put them in a bag and call it \(\phi\)

how many ways can arrange the objects \(\{\phi, ~P, ~R, ~M,~T,~T,~N,~S\}\) ?

=(8/2)!

12!/(12-5)! for i?

you mean 8!/2 ?

ya this one

Yes, next unpack the bag, how many ways can you arrange 5 vowels ?

can vowels be repeated

what letters do you have in the bag ?

aeiou

they all are distinct, how many ways can you arrange them ?

5!

Yes, so the final answer is ?

8!*5!/2

looks good!

ok the next one

separated than the first condition ?

P and S are at distance of 4 places always

First find the number of ways of placing \(P, S\) :
|dw:1439069671413:dw|

12*11

nope, try again

11

|dw:1439070002368:dw|

If one letter of {P, S} is at index 1, where can the other letter be ?

Yes, {P, S} can be mapped to {1, 6}

and you can swap the locations, so there are 7*2 = 14 different ways to arrange {P, S}

Next, arrange remaining 10 letters

10!/2

Yes, so total num of permutations = 14(10!/2)

if the restriction is not there, then there are 12!/2 different permutations
not 10!/2

ohk i see the difference :D

hmm this little silly doubts make my brain spin

its natural, my brain spins too :)

listing the possibilities and looking at the problem from alternative ways usually helps...

but in the way of learning its essential to get rid of the small doubts

yes thumbs up!

remembering school days where teachers used to run after students after student asked questions