In how numerous ways deserve to 6 in different way colored increased bushes be planted in a heat that only has actually room for 4 bushes?

Explanation:

There room 6 means to choose the an initial rose bush, 5 methods to pick the second, 4 methods to choose the third, and 3 methods to choose the fourth. In total there room 6 * 5 * 4 * 3 = 360 methods to kinds the rose bushes.

You are watching: How many different arrangements of 5 letters can be made from the 26 letters of the alphabet?

Explanation:

We have to arrange the 5 balls in 5 positions: _ _ _ _ _. The an initial position have the right to be bring away by any type of of the 5 balls. Climate there are 4 balls left to fill the second position, and also so on. Because of this the variety of arrangements is 5! = 5 * 4 * 3 * 2 * 1 = 120.

Daisy wants to arrange 4 vases in a row outside of she garden. She has eight vases to pick from. How many vase arrangements have the right to she make?

Explanation:

For this problem, because the stimulate of the vases problem (red blue yellow is different than blue red yellow), we"re dealing with permutations.

With

potential options, the total variety of possible permutations(order matters) is:

What is the minimum amount of handshakes that deserve to occur amongst fifteen human being in a meeting, if each human only shakes each various other person"s hand once?

Explanation:

This is a combination problem the the form “15 select 2” due to the fact that the sets of handshakes carry out not matter in order. (That is, “A shakes B’s hand” is the same as “B shakes A’s hand.”) utilizing the typical formula us get: 15!/((15 – 2)! * 2!) = 15!/(13! * 2!) = (15 * 14)/2 = 15 * 7 = 105.

There room 20 world eligible for town council, which has actually three chosen members.

Quantity A

The number of possible combine of council members, presuming no differentiation among office-holders.

Quantity B

The number of possible combine of board of directors members, given that the council has actually a president, vice president, and also treasurer.

Explanation:

This is a issue of permutations and combinations. You could solve this utilizing the proper formulas, yet it is always the situation that you deserve to make more permutations 보다 combinations because that all teams of size greater than one because the bespeak of choice matters; therefore, there is no doing the math, you understand that B have to be the answer.

Joe has a collection of 10 books that he hasn"t however read. If that takes 3 of them on vacation, exactly how many feasible sets of books can the take?

Explanation:

He can choose from 10, climate 9, climate 8 books, but since order walk not matter we have to divide through 3 factorial

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

How plenty of different license passwords deserve to one make if stated password should contain exactly 6 characters, two of i beg your pardon are distinctive numbers, an additional of which have to be one upper-case letter, and also the remaining 3 deserve to be any kind of digit or letter (upper- or lower-case) such that there space no repetitions of any type of characters in the password?

Explanation:

Begin through considering the 3 "hard and also fast conditions" - the digits and also the one upper-case letter. Because that the an initial number, girlfriend will have 10 choices and also for the 2nd 9 (since you can not repeat). Because that the captial letter, you have 26 choices. Therefore far, your password has actually 10 * 9 * 26 feasible combinations.

Now, provided your continuing to be options, you have 8 digits, 25 upper-case letters, and 26 lower-case letters (i.e. 59 possible choices). Because you cannot repeat, you will thus have actually for your remaining options 59, 58, and also 57 possibilities.

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

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### Example inquiry #1 : exactly how To discover The greatest Or Least number of Combinations

In how plenty of different orders deserve to 8 players sit ~ above the basketball bench?

40,320
36
362,880
3,040
15,700
Explanation:

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 also so on. Thus, 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 or 8! = 40, 320.

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### Example question #2 : how To discover The greatest Or Least variety of Combinations

There space 300 human being at a networking meeting. How plenty of different handshakes are feasible among this group?

None that the various other answers

89,700

45,000

300!

44,850

44,850

Explanation:

Since the order of persons shaking hands does not matter, this is a instance of computing combinations. (i.e. The is the same thing for human 1 come shake hands with human being 2 as it is for human being 2 come shake hand with human 1.)

According to our combinations formula, we have:

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

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### Example question #3 : how To uncover The greatest Or Least variety of Combinations

What is the number of possible 4-letter indigenous that deserve to be made from the 26 letter in the alphabet, where all 4 letters must be different?

Assume non-sensical words count, i.e. "dnts" would certainly count as a 4-letter word for our purposes.

760,400

100,000

530,600

250,000

358,800

358,800

Explanation:

This is a permutation that 26 letter taken 4 at a time. Come compute this us multiply 26 * 25 * 24 * 23 = 358,800.

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