Mastering Permutations and Combinations

Front

What is a permutation?

Back

A permutation is an arrangement of objects in a specific order. The number of permutations of n distinct objects is given by n! (n factorial).

Front

What is a combination?

Back

A combination is a selection of objects without regard to the order. The number of combinations of n objects taken r at a time is given by \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

Front

How do you calculate the number of ways to arrange r objects from n distinct objects?

Back

The number of ways to arrange r objects from n distinct objects is given by the formula: \( P(n, r) = \frac{n!}{(n-r)!} \).

Front

What is the formula for combinations?

Back

The formula for combinations is: \( C(n, r) = \frac{n!}{r!(n-r)!} \).

Front

If a password consists of 3 letters chosen from the alphabet with repetition allowed, how many different passwords can be formed?

Back

The number of different passwords is given by: \( 26^3 = 17,576 \).

Front

What is the probability of getting exactly two heads in four flips of a fair coin?

Back

The probability is given by: \( P(X=2) = \binom{4}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^2 = \frac{3}{8} \).

Front

How many ways can a 4-member relay team be selected from 8 athletes?

Back

The number of ways to select the team is given by: \( C(8, 4) = 70 \).

Front

What is the probability of selecting 3 red balls from a bag containing 5 red and 7 blue balls without replacement?

Back

The probability is calculated as: \( P = \frac{C(5, 3) \cdot C(7, 0)}{C(12, 3)} = \frac{10}{99} \).

Front

What is the difference between permutations and combinations?

Back

Permutations consider the order of selection, while combinations do not. For example, ABC is different from ACB in permutations but the same in combinations.

Front

What is the factorial of a number n?

Back

The factorial of n, denoted as n!, is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.