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 \( C(n, r) = \frac{n!}{r!(n-r)!} \).

Front

How do you calculate the probability of an event?

Back

The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes: \( P(E) = \frac{n(E)}{n(S)} \).

Front

What is the formula for calculating permutations of n objects taken r at a time?

Back

The formula is \( P(n, r) = \frac{n!}{(n-r)!} \).

Front

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

Back

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

Front

What is the probability of getting exactly k successes in n trials?

Back

The probability is given by the binomial formula: \( P(X = k) = C(n, k) p^k (1-p)^{n-k} \), where p is the probability of success.

Front

How many ways can you arrange 3 letters from the word 'CAT'?

Back

The number of arrangements is 3! = 6.

Front

If you have 5 red balls and 7 blue balls, how do you calculate the probability of drawing 3 red balls without replacement?

Back

Use the formula: \( P = \frac{C(5, 3)}{C(12, 3)} = \frac{10}{220} = \frac{1}{22} \).

Front

What is the total number of outcomes when flipping a coin 4 times?

Back

The total number of outcomes is \( 2^4 = 16 \).

Front

How many different passwords can be created using 3 distinct letters from the alphabet with repetition allowed?

Back

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