Back to Exercise: Use permutations and combinations to compute probabilities

Exercises: Use Permutations and Combinations to Compute Probabilities

Work through each section in order. Before computing a count, ask the deciding question: "If I swap two of my chosen items, is it a DIFFERENT outcome?" If yes, order matters — use a permutation, nPr = n!/(n - r)!. If no, order does not matter — use a combination, nCr = nPr / r! = n!/(r!(n - r)!). For probabilities, count the favorable and the total the SAME way (both ordered or both unordered).

Grade 11·20 problems·~35 min·Common Core Math - HS Statistics and Probability·group·hss-cp-b-9

Work through problems with immediate feedback

Warm-Up: Counting Principle, Factorials, and Order

These problems review the counting building blocks.

A deli offers 3 kinds of bread and 5 kinds of filling. A sandwich is one bread and one filling. By the Fundamental Counting Principle, how many different sandwiches are possible?

$3 \times 5 = 15$

$3 + 5 = 8$

$5 - 3 = 2$

$3! = 6$

Compute the factorial $5!$ ($5! = 5 \times 4 \times 3 \times 2 \times 1$).

You choose 3 people from a group. In which situation does ORDER MATTER (so swapping two chosen people gives a different outcome)?

Choosing a president, then a vice-president, then a treasurer.

Choosing a 3-person committee with no titles.

Choosing 3 people to all receive the same prize.

Choosing 3 people to form a study group.

Fluency Practice

Compute each count. Cancel factorials before multiplying out.

A 4-character password uses one digit (0-9) for each character, repeats allowed. By the counting principle, how many passwords are possible? (Enter the number.)

Compute the permutation $_8P_3 = \dfrac{8!}{(8-3)!} = \dfrac{8!}{5!}$ .

Compute the combination $_8C_3 = \dfrac{_8P_3}{3!} = \dfrac{8!}{3!\,(8-3)!}$ .

A shelf will hold 4 of 7 distinct books arranged left to right. How many arrangements are possible? Compute $_7P_4$ .

How many different 3-topping pizzas can be made from 6 available toppings, if no topping is repeated and order does not matter? Compute $_6C_3$ .

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