In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

In the word 'CORPORATION', we treat the vowels OOAIO as one letter.

Thus, we have CRPRTN (OOAIO).

This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different.

Number of ways arranging these letters =7!/2!= 2520.

Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in 3!/5!= 20 ways.

Required number of ways = (2520 x 20) = 50400.

A team of 6 members has to be selected from the 10 players. This can be done in 10C6 or 210 ways.

Now, the captain can be selected from these 6 players in 6 ways.
Therefore, total ways the selection can be made is 210×6= 1260

When R and W are the first and last letters of all the words then we can arrange them in 5!ways. Similarly When W and R are the first and last letters of the words then the remaining letters can be arrange in 5! ways.

Thus the total number of permutations = 2 x 5!  = 2 x 120 = 240

The first, the seventh, the ninth and the tenth letters of the word RECREATIONAL are R, T, O and N respectively. Meaningful word from these letters is only TORN. The third letter of the word is ?R?.

The given digits are 1, 2, 3, 5, 7, 9  

Number of even numbers = ?P? = 60.

Three digit number will have unit?s, ten?s and hundred?s place.

Out of 5 given digits any one can take the unit?s place.

This can be done in 5 ways. ...              (i)

After filling the unit?s place, any of the four remaining digits can take the ten?s place.

This can be done in 4 ways. ...              (ii)

After filling in ten?s place, hundred?s place can be filled from any of the three remaining digits.

This can be done in 3 ways. ... (iii) 

So,by counting principle, the number of 3 digit numbers = 5x 4 x 3 = 60

P(4,3)= P 3 4= 24

6P4 = 6! / (6-4)! = 360

C 5 20 = 15504

Since it does not matter what order the committee members are chosen in, the combination formula is used.

Committees are always a combination unless the problem states that someone like a president has higher hierarchy over another person. If the committee is ordered, then it is a permutation.

C(17,7)= 19,448