A bag contains 5 green and 7 red jellybeans. How many ways can 5 jellybeans be withdrawn from the bag so that the number of green ones withdrawn will be less than 2?
This can be calculated as the sum of two combinations; a draw with zero greens, and a draw with one green.
Choose 5 of 7 reds:
\(C(7,5) = \frac{7!}{5!(7-5)!}\) = 21
Choose 1 of 5 greens, and choose 4 of 7 reds:
\(C(5,1) \times C(7,4) = \frac{5!}{1!4!} \times \frac{7!}{4!3!} = 5 \times 35 = 175\)
21 + 175 = 196
A certain congressional committee consists of 14 senators and 13 representatives. How many ways can a subcommittee of 5 be formed if at least 4 of the members must be representatives?
This can be calculated as the sum of two combinations; a committee with five representatives, and a committee with four representatives.
Choose 5 of 13 representatives:
\(C(13,5) = \frac{13!}{5!(13-5)!}\) = 1287
Choose 1 of 14 senators, and 4 of 13 representatives:
\(C(14,1) \times C(13,4) = \frac{14!}{13!} \times \frac{13!}{4!9!}\) = 1.001^{4} = 10,010
If a coin is tossed 5 times, and then a standard six-sided die is rolled 2 times, and finally a group of three cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?
5 Coin Toss: \(2^5 = 32\)
2 Die Rolls: \(6^2 = 36\)
3 Card Draws: \(P(52,3) = \frac{52!}{52-3} = 52 \times 51 \times 50 = 132,600\)
32 + 36 + 132,600 = 132,668
3 cards are drawn from a standard deck without replacement. What is the probability that at least one of the cards drawn is a 3? Express your answer as a fraction or a decimal number rounded to four decimal places.
All possible draws \(C(52,3) = \frac{52!}{3!49!}\) = 2.21^{4}
Draws without 3 \(C(48,3) = \frac{48}{3!45!}\) = 1.7296^{4}
P(at least one 3) = \(1 - \frac{17296}{22100}\) = 0.2174 = 21.74%
Lorenzo is picking out some movies to rent, and he is primarily interested in documentaries and mysteries. He has narrowed down his selections to 17 documentaries and 14 mysteries.
Step 1. How many different combinations of 5 movies can he rent?
The combinations of 5 of 31 total movies is:
\(C(31,5) = \frac{31!}{5!26!}\) = 1.69911^{5} = 169,911
Step 2. How many different combinations of 5 movies can he rent if he wants at least one mystery?
This can be calculated as the product of two combinations; 1 of 14 mysteries, and 4 of 30 remaining selections:
\(C(14,1) = \frac{14!}{13!}\) = 14
\(C(30,4) = \frac{30!}{4!26!}\) = 2.7405^{4}
14 * 27405 = 383,670
In choosing what music to play at a charity fund raising event, Cory needs to have an equal number of symphonies from Brahms, Haydn, and Mendelssohn. If he is setting up a schedule of the 9 symphonies to be played, and he has 4 Brahms, 104 Haydn, and 17 Mendelssohn symphonies from which to choose, how many different schedules are possible? Express your answer in scientific notation rounding to the hundredths place.
This can be calculated as the product of three combinations; 3 of 4 Brahms, 3 of 104 Haydn, and 3 of 17 Mendelssohn:
\(C(4,3) = \frac{4!}{3!}\) = 4
\(C(103,3) = \frac{103!}{3!100!}\) = 1.76851^{5}
\(C(17,3) = \frac{17!}{3!14!}\) = 680
\(4 \times 176851 \times 680\) = 4.81 x 10^8
An English teacher needs to pick 13 books to put on his reading list for the next school year, and he needs to plan the order in which they should be read. He has narrowed down his choices to 6 novels, 6 plays, 7 poetry books, and 5 nonfiction books.
Step 1. If he wants to include no more than 4 nonfiction books, how many different reading schedules are possible? Express your answer in scientific notation rounding to the hundredths place.
This can be calculated as the sum of several combinatorial sequences, choosing 0-4 of 5 nonfiction books and 9-13 of the 19 remaining selections:
0 Nonfiction, 13 of 19 others
\(C(19,13) = \frac{19!}{13!6!}\) = 1.4065229^{10} +
1 of 5 Nonfiction, 12 of 19 others
\(C(5,1) \times C(19,12) = \frac{5!}{1!4!} \times \frac{19!}{12!7!}\) = 6.3996791^{12} +
2 of 5 Nonfiction, 11 of 19 others
\(C(5,2) \times C(19,11) = \frac{5!}{2!3!} \times \frac{19!}{11!8!}\) = 4.4234582^{16} +
3 of 5 Nonfiction, 10 of 19 others
\(C(5,3) \times C(19,10) = \frac{5!}{3!2!} \times \frac{19!}{10!9!}\) = 4.865804^{17} +
4 of 5 Nonfiction, 9 of 19 others
\(C(5,4) \times C(19,9) = \frac{5!}{4!1!} \times \frac{19!}{9!10!}\) = 6.082255^{18}
= 6.6130764^{18} = 6.61 x 10^18
Step 2. If he wants to include all 6 plays, how many different reading schedules are possible? Express your answer in scientific notation rounding to the hundredths place.
This would be the product of two combinations; 6 of 6 plays (equaling 1), and 7 of the 18 remaining titles.
\(C(6,6) \times C(18,7) = \frac{6!}{6!0!} \times \frac{18!}{7!11!}\) = 5.07068^{19} = 5.07 x 10^19
Zane is planting trees along his driveway, and he has 5 sycamores and 5 cypress trees to plant in one row. What is the probability that he randomly plants the trees so that all 5 sycamores are next to each other and all 5 cypress trees are next to each other? Express your answer as a fraction or a decimal number rounded to four decimal places.
There are 5! ways to combine each group of sycamore and cypress trees, and two possible ways to order these groups in a single row (e.g. sycamores first or last.) By dividing this product into the 10! possible combinations of trees without any restrictions, we get the probability:
\(\frac{5!5!2}{10!} = \frac{120^2 \times 2}{3,628,800}\) = 0.0079365 = 0.7936%
If you draw a queen or lower from a standard deck of cards, I will pay you $4. If not, you pay me $16. (Aces are considered the highest card in the deck.)
Step 1. Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
There are 8 cards with the rank of King or Ace in the deck, and 44 cards below that rank. The expected value of any given draw is:
$(P(Q-) $4) - (P(K+) $16) = ( $4) - ( $16) = $3.38 - $2.46 $ = $0.92
Step 2. If you played this game 833 times how much would you expect to win or lose? Round your answer to two decimal places. Losses must be expressed as negative values.
\(\$0.92 \times 833\) = $766.36