Probabilities

ballin_sensation · #1 $Old$ October 14th, 2009, 09:07 AM

How many ways can a student mark her answer sheet for a multiple
choice test if the first 3 questions have four choices for each question,
and the following 5 questions have three choices for each question and
only one choice is selected for each question?

A SUPERFECTA is a four horse bet, where one must pick the first,
second, third and fourth place horses in the exact order in which they
finish. In an six horse race, how many SUPERFECTA bets are possible?

Thanks for the help!

Soroban · #2 $Old$ October 14th, 2009, 09:31 AM

Hello, ballin_sensation!

Quote:

How many ways can a student mark her answer sheet for a multiple-choice test
if the first 3 questions have four choices for each question,
and the following 5 questions have three choices for each question
and only one choice is selected for each question?

In Part 1, she has a choice of 4 answers for each of the 3 questions.
. . She can mark Part 1 in: . $4^3 = 64$ ways.

In Part 2, she has a choice of 3 answers for each of the 5 questions.
. . She can mark Part 2 in: . $3^5 = 243$ ways.

Therefore, she can mark the test in: . $64 \times 243 \:=\:15,\!552$ ways.

Quote:

A Superfecta is a four-horse bet, where one must pick the first, second,
third and fourth place horses in the exact order in which they finish.
In an six-horse race, how many Superfecta bets are possible?

$\begin{array}{c}\text{There are 6 choices for the 1st place horse.} \\ \text{There are 5 choices for the 2nd place horse.} \\ \text{There are 4 choices for the 3rd place horse.} \\ \text{There are 3 choices for the 4th place horse.} \end{array}$

Therefore, there are: . $6\cdot5\cdot4\cdot3 \:=\:360$ possible Superfecta bets.