Combinatorics
Math ⇒ Statistics and Probability
Combinatorics starts at 9 and continues till grade 12.
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See sample questions for grade 12
A bag contains 4 red, 5 blue, and 3 green balls. How many ways can 3 balls be drawn if at least one must be green?
A class has 12 students. In how many ways can a president, vice-president, and secretary be chosen?
A committee of 4 is to be formed from 7 men and 5 women. How many ways can the committee be formed if it must contain exactly 2 women?
A password consists of 2 letters followed by 3 digits. How many different passwords can be formed if repetition is allowed?
A bag contains 4 red, 5 blue, and 3 green balls. How many ways can 3 balls be drawn if at least one must be green?
A class has 12 students. In how many ways can a president, vice-president, and secretary be chosen?
A committee of 4 is to be formed from 7 men and 5 women. How many ways can the committee be formed if it must contain exactly 2 women?
A password consists of 2 letters followed by 3 digits. How many different passwords can be formed if repetition is allowed?
Which of the following is NOT a property of binomial coefficients?
(1) \( \binom{n}{0} = 1 \)
(2) \( \binom{n}{n} = 1 \)
(3) \( \binom{n}{r} = \binom{n}{n-r} \)
(4) \( \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1} \)
(5) \( \binom{n}{r} = n \cdot \binom{n-1}{r-1} \)
Which of the following is the correct expansion for (x + y)^4?
(1) x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
(2) x^4 + 4x^2y^2 + y^4
(3) x^4 + 4xy^3 + y^4
(4) x^4 + 6x^2y^2 + y^4
Which of the following is the correct formula for the number of multisets of size r from n types?
(1) n^r
(2) \( \binom{n+r-1}{r} \)
(3) \( \binom{n}{r} \)
(4) r^n
Which of the following is the correct formula for the number of permutations of n objects taken r at a time?
(1) n! / (n - r)!
(2) n! / r!
(3) n^r
(4) r! / (n - r)!
Fill in the blank: The number of non-negative integer solutions to the equation x + y + z = 7 is ________.
Fill in the blank: The number of ways to arrange n distinct objects in a circle is ________.
Fill in the blank: The number of ways to distribute n identical balls into r distinct boxes is ________.
Fill in the blank: The number of ways to select at least one object from n distinct objects is ________.
True or False: In permutations, the order of selection is important.
True or False: The number of combinations of n objects taken r at a time is always less than or equal to the number of permutations of n objects taken r at a time.
True or False: The number of ways to arrange n objects in a circle is n!.
True or False: The number of ways to arrange n objects in a line is n!.
