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).
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?
$3! = 6$
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 .
Compute the combination .
A shelf will hold 4 of 7 distinct books arranged left to right. How many arrangements are possible? Compute .
How many different 3-topping pizzas can be made from 6 available toppings, if no topping is repeated and order does not matter? Compute .
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