Permutation Combination Problem

i think the answer should be 5C4*21C13*17!

oh yes it is 17!.. how did you get that? any explanation will be of great help

Also what will be the answer for -

How many 17-letter words can be formed using the 26 lower-case letters of the alphabet if:Repetition of letters is not allowed?

That would be 26P17

ladhanivishal wrote:oh yes it is 17!.. how did you get that? any explanation will be of great help

because those words can be arranged in 17! ways

In the first case : "How many 17-letter words can be formed using the 26 lower-case letters of the alphabet if:Each word contains exactly 4 vowels, with repetition of letters not allowed "

The Answer is 5C4*21C13*17!

whereas in the second case :

"How many 17-letter words can be formed using the 26 lower-case letters of the alphabet if:Repetition of letters is not allowed?"

Answer is 26P17

How is that the first one is a combination and the second one a permutation?

Please explain..

ladhanivishal wrote:In the first case : "How many 17-letter words can be formed using the 26 lower-case letters of the alphabet if:Each word contains exactly 4 vowels, with repetition of letters not allowed "

The Answer is 5C4*21C13*17!

whereas in the second case :

"How many 17-letter words can be formed using the 26 lower-case letters of the alphabet if:Repetition of letters is not allowed?"

Answer is 26P17

How is that the first one is a combination and the second one a permutation?

Please explain..

buddy, isn't 26P17 same as 26C17*17!

The second answer can also be read as 26C17*!17. This is same as 26P17.

The theme of both the answers is same. First select the number of letters from the given set in whatever ways they can be selected (5C4*21C13 or 26C17) and then arrange them (!17 ways for 17 letters of either question). We multiply them to arrive at the answers.

After removing 5 vowels from 26 Alphabets we have 21 consonants

i) Selecting 4 vowels out of 5= 5C4
ii)Selecting 13 consonants our of 21= 21C13

These 17 selected alphabets can arrange themselves in 17! ways.

So total number of ways- 5C4*21C13*17!

gotcha guys.. thanks!

couldn't be simply ?

Vowels= 5*4*3*2
Consonants= 21*20*19*18*17*16*15*14*13*12*11*10*9

ILULA08 wrote:After removing 5 vowels from 26 Alphabets we have 21 consonants

i) Selecting 4 vowels out of 5= 5C4
ii)Selecting 13 consonants our of 21= 21C13

These 17 selected alphabets can arrange themselves in 17! ways.

So total number of ways- 5C4*21C13*17!

What is derived if you don't multiply the extra 17! ???

I solved everything else, but left off 17! since I already thought I got the answer.

cata1yst wrote:

ILULA08 wrote:After removing 5 vowels from 26 Alphabets we have 21 consonants

i) Selecting 4 vowels out of 5= 5C4
ii)Selecting 13 consonants our of 21= 21C13

These 17 selected alphabets can arrange themselves in 17! ways.

So total number of ways- 5C4*21C13*17!

What is derived if you don't multiply the extra 17! ???

I solved everything else, but left off 17! since I already thought I got the answer.

Let's look at an analagous question to understand the difference:

1) Bob is having a party and is inviting a total of 17 friends. Bob will invite 4 men and 13 women to his party. If Bob has 5 male friends and 21 female friends, how many different groups of friends could he invite to his party?

In this question, we're selecting 4 out of 5 men and 13 out of 21 women, so the answer would be:

5C4 * 21C13

(remember, when we make MULTIPLE selections we MULTIPLY the results).

Next, let's add an extra dimension to the question:

2) Bob is hosting a party with 17 friends. At the party, all 17 guests will dance in a conga line (https://www.youtube.com/watch?v=zI5mfE9dhFU). In how many different ways can the guests be arranged for the dance?

In this question, we're arranging 17 people, so the answer would be:

17P17 = 17!

Now let's combine the two questions:

3. Bob is having a party and is inviting a total of 17 friends. Out of Bob's 5 male friends and 21 female friends, he will invite 4 men and 13 women to his party.

At Bob's party, all 17 of his friends will dance in a conga line. Out of Bob's total of 26 friends, how many different conga lines can be formed?

In this question, we're not only selecting 4 out of 5 and 13 out of 21, we're also arranging those that we select. So, our final answer is:

(# of selections) * (# of arrangements)

= (5C4*21C13) * (17!)