Math 2000 - Solutions 12
Fall 2001
Unit 7E 4. This is arrangement with repetition, and so we have $7^{10}= 282,475,249$ different ten-note tunes using the 7 available notes. 6. This is arrangement with repetition, and so we have $30^3=27,000$ different three-number combinations using the available 30 numbers. 8. This is a permutation, and so we have $12 \times 11 \times \dots\ \times 3 \times 2 \times 1 = 12! = 479,001,600$ different ways to schedule the 12 acts. 12. This is a combination, as we are only interested in the makeup of the hand, order plays no role. So we have ${}_{52}C_4$, which comes out to 270,725, different four-card hands, using the fifty-two available cards. 14. This is a permutation, and so we have $52 \times 51 \times 50 \times 49 \times 48= 311,875,200$ different five-card sequences, using the fifty-two available cards. 21. This is combination problem, as we are only interested in the makeup of the pizza, order plays no role. If Luigi uses $N$ different pizza toppings, then we have ${}_{N}C_3=84$ different three-topping pizzas. We wish to know $N$: some trial and error leads to the discovery that $N=9$. Similarly, if Ramona uses $M$ different pizza toppings, then we have ${}_{M}C_2=45$ different two-topping pizzas. We wish to know $M$: some trial and error leads to the discovery that $M=10$. Lotteries We have seen that lottery problems involve combinations. In the first lottery, there are ${}_{56}C_5=3,819,816$ different arrangements of the numbers, which means the probability of matching one of those numbers with one ticket is $1/3,819,816 = 2.6179 \times 10^{-7}$. For the second lottery, there are ${}_{40}C_6=3,838,380$ different arrangements of the numbers, which means the probability of matching one of those numbers with one ticket is $1/3,838,380 = 2.6053 \times 10^{-7}$. For the third lottery, there are ${}_{33}C_7=4,272,048$ different arrangements of the numbers, which means the probability of matching one of those numbers with one ticket is $1/4,272,048 = 2.3408 \times 10^{-7}$. We see that the first lottery has the greatest probability of winning because it has the fewest number of outcomes.