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Five points lie on a circle. How many chords can be drawn through them?
10
How many diagonals does a hexagon have?
There are five teams in a league. How many games are played if every team plays each other twice?
20
A team plays 15 games a season. In how many ways can it have 8 wins and 7 losses?
In how many different ways can a 4-child family have 2 boys and 2 girls?
6
A coin is tossed five times. In how many ways can it fall three heads and two tails?
The shopping area of a town is a square that is six blocks by six blocks. How many different routes can a taxi driver take to go from one corner of the shopping area to the opposite cater-corner?
924
If the shopping area in [link] has a rectangular form of 5 blocks by 3 blocks, then how many different routes can a taxi driver take to drive from one end of the shopping area to the opposite kitty corner end?
Following problems involve combinations from several different sets.
How many 5-people committees consisting of three boys and two girls can be chosen from a group of four boys and four girls?
24
A club has 4 men, 5 women, 8 boys and 10 girls as members. In how many ways can a group of 2 men, 3 women, 4 boys and 4 girls be chosen?
How many 4-people committees chosen from four men and six women will have at least three men?
25
A batch contains 10 transistors of which three are defective. If three are chosen, in how many ways can one get two defective?
In how many ways can five counters labeled A, B, C, D and E at a store be staffed by two men and three women chosen from a group of four men and six women?
14,400
How many 4-letter word sequences consisting of two vowels and two consonants can be made from the letters of the word PHOENIX if no letter is repeated?
Three marbles are chosen from an urn that contains 5 red, 4 white, and 3 blue marbles. How many samples of the following type are possible?
Two blue and one white.
All three of the same color.
None red.
Five coins are chosen from a bag that contains 4 dimes, 5 nickels, and 6 pennies. How many samples of five of the following type are possible?
No pennies.
Four of a kind.
Three of one kind and two of another kind.
Find the number of different ways to draw a 5-card hand from a deck to have the following combinations.
A heart flush(all hearts).
Two cards of one suit, and three of another suit.
Two cards of one value and three of another value.
Use the Binomial Theorem to do the following problems.
Expand ${\left(a+b\right)}^{5}$ .
Expand ${\left(a-b\right)}^{6}$ .
Expand ${\left(x-\mathrm{2y}\right)}^{5}$ .
Expand ${\left(\mathrm{2x}-\mathrm{3y}\right)}^{4}$ .
Find the third term of ${\left(\mathrm{2x}-\mathrm{3y}\right)}^{6}$ .
Find the sixth term of ${\left(\mathrm{5x}+y\right)}^{8}$ .
Find the coefficient of the ${x}^{3}{y}^{4}$ term in the expansion of ${\left(\mathrm{2x}+y\right)}^{7}$ .
280
Find the coefficient of the ${a}^{4}{b}^{6}$ term in the expansion of ${\left(\mathrm{3a}-b\right)}^{\text{10}}$ .
A coin is tossed 5 times, in how many ways is it possible to get three heads and two tails?
10
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