There are 6 ways to choose the first rose bush, 5 ways to choose the second, 4 ways to choose the third, and 3 ways to choose the fourth. In total there are 6 * 5 * 4 * 3 = 360 ways to arrange the rose bushes.

We need to arrange the 5 balls in 5 positions: _ _ _ _ _.  The first position can be taken by any of the 5 balls. Then there are 4 balls left to fill the second position, and so on.  Therefore the number of arrangements is 5! = 5 * 4 * 3 * 2 * 1 = 120.

For this problem, since the order of the vases matters (red blue yellow is different than blue red yellow), we're dealing with permutations.

With  selections made from  potential options, the total number of possible permutations(order matters) is:

This is a combination problem of the form “15 choose 2” because the sets of handshakes do not matter in order.  (That is, “A shakes B’s hand” is the same as “B shakes A’s hand.”)  Using the standard formula we get: 15!/((15 – 2)! * 2!) = 15!/(13! * 2!) = (15 * 14)/2 = 15 * 7 = 105.

This is a matter of permutations and combinations. You could solve this using the appropriate formulas, but it is always the case that you can make more permutations than combinations for all groups of size greater than one because the order of selection matters; therefore, without doing the math, you know that B must be the answer. 

He can choose from 10, then 9, then 8 books, but because order does not matter we need to divide by 3 factorial

 (10 * 9 * 8) ÷ (3 * 2 * 1) = 720/6 = 120

Begin by considering the three "hard and fast conditions" - the digits and the one upper-case letter.  For the first number, you will have 10 choices and for the second 9 (since you cannot repeat).  For the captial letter, you have 26 choices.  Thus far, your password has 10 * 9 * 26 possible combinations.

Now, given your remaining options, you have 8 digits, 25 upper-case letters, and 26 lower-case letters (i.e. 59 possible choices).  Since you cannot repeat, you will thus have for your remaining choices 59, 58, and 57 possibilities.

Putting all of this together, you have: 10 * 9 * 26 *59 * 58 * 57 or 456426360 choices.

Using the Fundamental Counting Principle, there would be 8 choices for the first player, 7 choices for the second player, 6 for the third, 5 for the fourth, and so on.  Thus, 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 or 8! = 40, 320.

Since the order of persons shaking hands does not matter, this is a case of computing combinations.  (i.e. It is the same thing for person 1 to shake hands with person 2 as it is for person 2 to shake hands with person 1.)

According to our combinations formula, we have:

300! / ((300-2)! * 2!) = 300! / (298! * 2) = 300 * 299 / 2 = 150 * 299 = 44,850 different handshakes

This is a permutation of 26 letters taken 4 at a time.  To compute this we multiply 26 * 25 * 24 * 23 = 358,800.