Factorial, Permutation & Combination Quiz
Grade 10 Level Practice Problems
Question 1: Evaluate the expression: 6!
Correct Answer: C
Solution: A factorial means multiplying a sequence of descending natural numbers down to 1.
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
Solution: A factorial means multiplying a sequence of descending natural numbers down to 1.
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
Question 2: Simplify the fraction: 8! / 6!
Correct Answer: D
Solution: Expand the larger factorial until it matches the smaller one, then cancel out common factors:
(8 × 7 × 6!) / 6! = 8 × 7 = 56
Solution: Expand the larger factorial until it matches the smaller one, then cancel out common factors:
(8 × 7 × 6!) / 6! = 8 × 7 = 56
Question 3: In how many different ways can a student council elect a President, Vice-President, and Secretary from a group of 7 candidates?
Correct Answer: A
Solution: Because the roles (positions) are distinct, order matters, making this a Permutation problem.
We use _7P_3 = 7! / (7-3)! = 7 × 6 × 5 = 210 ways.
Solution: Because the roles (positions) are distinct, order matters, making this a Permutation problem.
We use _7P_3 = 7! / (7-3)! = 7 × 6 × 5 = 210 ways.
Question 4: A student needs to choose 3 books to read from a summer reading list of 9 books. How many different selections can be made?
Correct Answer: B
Solution: The order in which the books are selected does not matter, so this is a Combination problem.
We use _9C_3 = 9! / (3!(9-3)!) = (9 × 8 × 7) / (3 × 2 × 1) = 504 / 6 = 84 ways.
Solution: The order in which the books are selected does not matter, so this is a Combination problem.
We use _9C_3 = 9! / (3!(9-3)!) = (9 × 8 × 7) / (3 × 2 × 1) = 504 / 6 = 84 ways.
Question 5: Which of the following situations represents a permutation?
Correct Answer: C
Solution: In a bank PIN, the arrangement order of the numbers changes the code (e.g., 1234 is different from 4321). All other options represent combinations because the order of selection changes nothing.
Solution: In a bank PIN, the arrangement order of the numbers changes the code (e.g., 1234 is different from 4321). All other options represent combinations because the order of selection changes nothing.
Question 6: Evaluate the permutation notation expression: _5P_2
Correct Answer: B
Solution: Using the permutation formula _nP_r = n! / (n-r)!:
_5P_2 = 5! / (5-2)! = 5! / 3! = 5 × 4 = 20.
Solution: Using the permutation formula _nP_r = n! / (n-r)!:
_5P_2 = 5! / (5-2)! = 5! / 3! = 5 × 4 = 20.
Question 7: Evaluate the combination notation expression: _6C_4
Correct Answer: A
Solution: Using the combination formula _nC_r = n! / (r!(n-r)!):
_6C_4 = 6! / (4! × 2!) = (6 × 5) / (2 × 1) = 30 / 2 = 15.
Tip: Note that _6C_4 is equal to _6C_2!
Solution: Using the combination formula _nC_r = n! / (r!(n-r)!):
_6C_4 = 6! / (4! × 2!) = (6 × 5) / (2 × 1) = 30 / 2 = 15.
Tip: Note that _6C_4 is equal to _6C_2!
Question 8: How many unique ways can the letters of the word "CANADA" be rearranged?
Correct Answer: C
Solution: This involves permutations with repeating items. "CANADA" has 6 total letters, where 'A' repeats 3 times.
Formula: n! / a! = 6! / 3! = (6 × 5 × 4 × 3!) / 3! = 6 × 5 × 4 = 120 unique ways.
Solution: This involves permutations with repeating items. "CANADA" has 6 total letters, where 'A' repeats 3 times.
Formula: n! / a! = 6! / 3! = (6 × 5 × 4 × 3!) / 3! = 6 × 5 × 4 = 120 unique ways.
Question 9: A hockey coach must choose 3 players out of 8 available players to take part in a post-game shootout. If the order of the shooters matters, how many ways can they be selected?
Correct Answer: B
Solution: Because the order of shooters determines who goes first, second, and third, it is a permutation.
_8P_3 = 8! / (8-3)! = 8 × 7 × 6 = 336 arrangements.
Solution: Because the order of shooters determines who goes first, second, and third, it is a permutation.
_8P_3 = 8! / (8-3)! = 8 × 7 × 6 = 336 arrangements.
Question 10: An ice cream shop offers 8 different flavors. An individual wants to buy a take-home tub containing exactly 2 different flavors mixed together. How many combinations are possible?
Correct Answer: C
Solution: Since the flavors are mixed in a single tub, order does not matter (Chocolate + Vanilla is the same as Vanilla + Chocolate).
_8C_2 = 8! / (2! × 6!) = (8 × 7) / 2 = 56 / 2 = 28 combinations.
Solution: Since the flavors are mixed in a single tub, order does not matter (Chocolate + Vanilla is the same as Vanilla + Chocolate).
_8C_2 = 8! / (2! × 6!) = (8 × 7) / 2 = 56 / 2 = 28 combinations.
Question 11: What is the value of 0!?
Correct Answer: B
Solution: By mathematical definition, 0! = 1. This convention ensures that formulas for permutations and combinations consistently yield correct values when selecting all items (e.g., _nP_n = n! / 0! = n!).
Solution: By mathematical definition, 0! = 1. This convention ensures that formulas for permutations and combinations consistently yield correct values when selecting all items (e.g., _nP_n = n! / 0! = n!).
Question 12: A classroom has 10 boys and 12 girls. A committee consisting of 2 boys and 2 girls needs to be formed. How many different committees can be made?
Correct Answer: A
Solution: Calculate the options for boys and girls independently using combinations, then apply the Fundamental Counting Principle to multiply them together:
Boys: _10C_2 = (10 × 9) / 2 = 45
Girls: _12C_2 = (12 × 11) / 2 = 66
Total Committees = 45 × 66 = 2,970.
Solution: Calculate the options for boys and girls independently using combinations, then apply the Fundamental Counting Principle to multiply them together:
Boys: _10C_2 = (10 × 9) / 2 = 45
Girls: _12C_2 = (12 × 11) / 2 = 66
Total Committees = 45 × 66 = 2,970.
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