The number of arrangements of 4 different digits taken 4 at a time is given by  4 P 4 = 4! = 24.All the four digits will occur equal number of times at each of the position,namely ones,tens,hundreds,thousands.

Thus,each digit will occur 24/4 = 6 times in each of the position.The sum of digits in one's position will be 6 x (1+3+5+7) = 96.Similar is the case in ten's,hundred's and thousand's places.

Therefore,the sum will be 96 + 96 x 10 + 96 x 100 + 96 x 100 = 106656

In a 3 digit number one?s place can be filled in 5 different ways with (0,2,4,6,8)

10?s place can be filled in 10 different ways

100?s place can be filled in 9 different ways

There fore total number of ways = 5X10X9 = 450

Given word is THERAPY.

Number of letters in the given word = 7

These 7 letters can be arranged in 7! ways.

Number of vowels in the given word = 2 (E, A)

The number of ways of arrangement in which vowels come together is 6! x 2! ways

Hence, the required number of ways can the letters of the word 'THERAPY' be arranged so that the vowels never come together = 7! - (6! x 2!) ways = 5040 - 1440 = 3600 ways.

NUMERICAL has 9 positions in which 2, 4, 6, 8 are even positions.

And it contains 5 consonents i.e, N, M, R, C & L. Hence this cannot be done as 5 letters cannot be placed in 4 positions.

Therefore, Can't be determined.

There are total 9 places out of which 4 are even and rest 5 places are odd.

4 women can be arranged at 4 even places in 4! ways.

and 5 men can be placed in remaining 5 places in 5! ways.

Hence, the required number of permutations  = 4! x 5! = 24 x 120 = 2880

First letter can be posted in 4 letter boxes in 4 ways. Similarly second letter can be posted in 4 letter boxes in 4 ways and so on.

Hence all the 5 letters can be posted in = 4 x 4 x 4 x 4 x 4 = 1024

The numbers 0,1,2,3,4,5,6,7,8,9 are 10 in number while preparing telephone numbers any number can be used any number of times.

This can be done in  10 5ways, but '0' is there

So, the numbers starting with '0' are to be excluded is  9 4 numbers.

 Total 5 digit telephone numbers =  10 5 - 9 4 = 3439

We have to find number of permutations of 4 objects out of 6 objects.

This number is 6 P 4= 360

Therefore, cards can be sent in 360 ways.

Choose 5 starters from a team of 12 players. Order is not important.

12 C 5= 729

Number of positive integral solutions =   n - 1 C r - 1 =  C 2 20 = 190