Problems

Worked counting and probability problems with model guidance

These pages bridge the gap between a raw formula and a real-world scenario, helping you decide whether the setup is counting, repeated-trial probability, or without-replacement probability.

Worked scenario

Choose 4 Fruits From 9

This is a plain combination problem because order does not matter and each fruit can be used once.

Read the worked problem

Worked scenario

Arrange 6 Books On a Shelf

This is a factorial or full permutation problem because all 6 distinct books are arranged in a row.

Read the worked problem

Worked scenario

Draw an Ace From a Standard Deck

This is a basic probability problem because you can count the favourable outcomes directly and divide by the full number of equally likely single-card outcomes.

Read the worked problem

Worked scenario

Get Exactly 3 Heads in 5 Coin Flips

This is a binomial probability problem because the flips are independent, each flip has the same probability of heads, and the question asks for an exact success count.

Read the worked problem

Worked scenario

Get At Least 1 Head in 4 Coin Flips

This is a binomial probability problem because the flips are independent repeated trials with the same success probability, and the question asks for a success-count range.

Read the worked problem

Worked scenario

Draw Exactly 2 Aces in 5 Cards

This is a hypergeometric probability problem because the cards are drawn without replacement from a fixed deck, so each draw changes the composition of the remaining deck. If you draw an ace first there would be one less to draw the second time. On every sequential draw there is one less card to draw from, changing the odds.

Read the worked problem

Worked scenario

Choose 5 Finalists From 12 Applicants

This is a plain combination problem because the panel is forming one group of finalists, order does not matter, and each applicant can be selected at most once.

Read the worked problem

Worked scenario

Seat 8 Guests Around a Round Table

This is a circular permutation problem because the order around the table matters, but rotating the same full seating does not create a new arrangement.

Read the worked problem

Worked scenario

Build a 3 Scoop Dessert From 5 Flavors

This is combination with repetition because the order of scoops does not matter, but the same flavor may appear more than once.

Read the worked problem

Worked scenario

Seat 7 Guests Around a Round Table

This is a circular permutation problem because rotating the full table does not create a new arrangement.

Read the worked problem

Worked scenario

Create 4 Digit Room Codes

This is permutation with repetition because the code positions are ordered and every position can reuse any symbol.

Read the worked problem

Worked scenario

Arrange the Word APPLE

This is a multiset permutation problem because the P appears twice, so duplicate swaps must be removed.

Read the worked problem