What is the definition of a permutation?

A permutation is an arrangement of objects in a specific order. The number of permutations of n distinct objects taken r at a time is calculated using the formula 
P(n, r) = n! / (n - r)! 
For Example, 
the number of ways to arrange 3 books from a set of 5 is P(5, 3) = 60.

What is the formula for the number of combinations of n distinct objects taken r at a time?

The formula for combinations is 
C(n, r) = n! / [r!(n - r)!],
where n is the total number of objects, and r is the number of objects being chosen. 
For Example, 
C(10, 3) calculates to 120.

If you have 7 different shirts and want to choose 4 to wear, how many different combinations can you make?

Using the combination formula:  Thus, there are 35 ways to choose the shirts.

How many ways can you arrange the letters in the word 'APPLE'?

Since 'APPLE' has 5 letters with the letter 'P' repeated twice, the arrangements can be calculated using the formula for arrangements of a multiset: 

5! / (2!) = 60. 

Therefore, there are 60 distinct arrangements.

A basketball team has 12 players. How many ways can a coach choose 5 players to start the game?

Using the combination formula: 

How many different ways can you arrange 5 books on a shelf?

The number of ways to arrange 5 distinct books is given by the permutation of all 5 books: P(5, 5) = 5! = 120. Therefore, there are 120 different arrangements.

If a committee of 4 is to be formed from 9 members, how many different committees can be formed?

Using the combination formula:
Thus, there are 126 ways to form the committee.

How many ways can you arrange the letters in the word 'MISSISSIPPI'?

The total arrangements of the letters in 'MISSISSIPPI' can be calculated using: 
= 11! / (4! × 4! × 2!) 
= 34650. 
Therefore, there are 34650 distinct arrangements.

What is a useful tip for solving permutation and combination problems quickly?

A useful tip is to identify whether the order of selection matters (permutation) or not (combination). Also, for combinations, remember C(n, r) = C(n, n - r) which can simplify calculations.

In how many different ways can 10 students be seated in a row of 10 chairs?

The number of ways to arrange 10 students in 10 chairs is 
P(10, 10) = 10!
 = 3628800
Therefore, there are 3628800 arrangements.

How many ways can you select a president, vice president, and secretary from a group of 5 people?

Since the order matters, this is a permutation problem: 
P(5, 3) = 5! / (5 - 3)! 
= 5! / 2! 
= 120. 
Thus, there are 120 ways to select the officers.

How many ways can you select 4 fruits from 10 different fruits if the selection order does not matter?

(Hint: Think about using the combination formula)

Using the combination formula: 
C(10, 4) = 10! / [4!(10 - 4)!] = 10! / (4! × 6!) 
= 210 
Thus, there are 210 ways to select the fruits.