Permutation and combination problem - word arrangement

Question

This is a question of permutation and combination. Q. How many words can be formed from the word "LUCKNOW" when

i) No restriction is there

ii) L is the first letter of the word

iii) All the vowels are together

iv) L always occurs before U

v) L always occurs before U and U always occurs before W

I have managed to solve i), ii) and iii), but no. iv) and v) I can't solve. Please help me with explanation to solve it.

Answer 1

i) All are distinct letters so, and there are 7 letters, all permutations will work which is equal to $7!$.

ii) L is first letter means, the rest letters are only allowed to be permuted, so 6 remaining letters so answer is all permutations of 6 letters i.e. $6!$.

iii) All vowels are together, you have for vowels here U and O, so in a word they always come together (As a block) and in 2 ways either UO or OU, all the words with UO is permutation of 5 letters and this one block, i.e. $(5+1)!$ and similarly for the other block, hence you have $2\times 6!$ in total.

iv) Note given a word, L either comes before U or after U. But suppose you have a word with U first and L coming afterwards, then if you switch the positions you have a corresponding word with L first and U coming afterwards, so they come in pairs, so they partition all words into two equal sets, so answer for this is $(7!)/2$.

v) Using the same logic as before, in a word where ..L..U..W.., is needed, if the word contains in any other order then just interchange places to get teh required order. So here we have more possibilities i.e. $3! = 6$ ways, they come in a set of 6, hence the partion is into 6 equal size sets hence the answer will be $(7!)/6$.

*In the above problem, I have assumed that I have to use all the letters to form the word, but if that is not the case, the values may vary but the logic still works.

Answer 2

Updated

For v) There are 5 positions for L (first 5 positions) The possible permutations of LUW for the first position of L equate to the 5th triangular number since there are 5+4+3+2+1 = 15 permutations of UW

For the second position of L we get the 4th triangular number = 10 and so on

Giving a total of 15 + 10 + 6 + 3 + 1 = 35 permutations for LUW for each of which there are 4! ways of arranging the other 4 letters.

So I make that 35*4! Which agrees with the answer from DiffeoR but may offer another way of looking at it.