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15 problems match “combinations”. Clear

Matches in a round-robin group ●○○○○ 10 ptsA group has four teams, and each team plays every other team in the group exactly once. How many matches are played in the group?
Coordinates of a vector in R^4 ●○○○○ 10 ptsLet $a_1 = [1, 1, 0, 0]$ , $a_2 = [0, 1, 1, 0]$ and $a_3 = [0, 0, 1, 1]$ . Find the coefficients $\alpha_1, \alpha_2, \alpha_3$ such that $v = [1, 3, 5, 3] = \alpha_1 a_1 + \alpha_2 a_2 + \alpha_3 a_3$ , that is, find the coordinate representation $[\alpha_1, \alpha_2, \alpha_3]$ of $v$ .
Choosing two balls from twenty ●○○○○ 10 ptsTwenty balls numbered 1 through 20 are in a bin. In how many ways can 2 balls be selected if the order in which they are drawn does not matter?
Removing an element to raise probability ●●○○○ 20 ptsTwo different numbers are randomly selected from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ . The probability that their sum is 12 would be greater if the number $n$ had first been removed from $S$ . What is the value of $n$ ?
Equal votes in a village election ●●○○○ 20 ptsA small village has $n$ people. During their yearly elections, every group of three people comes up to a stage and, as a group, casts a single vote for someone in the village to be the new leader. After every possible group of three people has voted, it turns out that every person in the village received exactly the same number of votes. If $10 \leq n \leq 100$ , how many possible values of $n$ are there?
Prime sum of white and red balls ●●○○○ 20 ptsA bag contains 6 red balls and 8 white balls. Five balls are taken at random and placed in box $A$ , and the remaining 9 balls are placed in box $B$ . Find the probability that the sum of the number of white balls in box $A$ and the number of red balls in box $B$ is not a prime number. Round to four decimal places.
Area of a parallelogram from vector combinations ●●○○○ 20 ptsCalculate the area of the parallelogram constructed on the vectors $\vec{a}=3\vec{p}+2\vec{q}$ and $\vec{b}=2\vec{p}-\vec{q}$ , given that $|\vec{p}|=4$ , $|\vec{q}|=3$ and the angle between $\vec{p}$ and $\vec{q}$ is $\frac{3\pi}{4}$ . Round to two decimal places.
Decomposing a vector in a basis ●●○○○ 20 ptsWrite the vector $x=(-5, -5, 5)$ as a linear combination of $p=(-2, 0, 1)$ , $q=(1, 3, -1)$ and $r=(0, 4, 1)$ , that is, find $\alpha$ , $\beta$ , $\gamma$ with $x=\alpha p+\beta q+\gamma r$ .
Coordinates in a non-standard basis ●●○○○ 20 ptsLet $v = [6, 3, 2, 5]$ , $a_1 = [2, 2, 2, 3]$ , $a_2 = [1, 0, -1, 0]$ and $a_3 = [0, 1, 0, 1]$ . Find the coordinates $[\alpha_1, \alpha_2, \alpha_3]$ of $v$ in terms of $a_1, a_2, a_3$ , that is, the coefficients with $v = \alpha_1 a_1 + \alpha_2 a_2 + \alpha_3 a_3$ .
Each player gets one ace ●●●○○ 50 ptsA deck of 52 cards containing four aces is shuffled thoroughly and the cards are dealt among four players so that each player receives 13 cards. Find the probability that each player receives exactly one ace. Give the answer as a percentage rounded to one decimal place.