How many ways can 6 books be arranged on a shelf?
There are 720 ways to arrange 6 books on a shelf.
We can use the formula for permutations of n objects taken r at a time:
P(n,r) = n! / (n-r)!
In this case, we want to arrange 6 books (n) on a shelf, which means we are taking all 6 books (r = 6).
P(6,6) = 6! / (6-6)!
P(6,6) = 6! / 0!
P(6,6) = 6 x 5 x 4 x 3 x 2 x 1
P(6,6) = 720
Therefore, there are 720 ways to arrange 6 books on a shelf.
There are 8 girls competing. How many ways can they win 1st, 2nd, and 3rd place
There are 8 girls competing, and we need to find the number of ways in which we can select the 1st, 2nd, and 3rd places.
There are 8 choices for the first place, since any one of the 8 girls can win. After a girl wins 1st place, there are 7 girls left to choose from for the 2nd place. Finally, there are 6 girls left for the 3rd place.
Using the multiplication principle, we can multiply these numbers to get the total number of ways in which we can select the 1st, 2nd, and 3rd places:
8 x 7 x 6 = 336
Therefore, there are 336 ways to win 1st, 2nd, and 3rd place among 8 competing girls.
how many ways can a president and vice president be elected in a class of 12
To elect a president and vice president in a class of 12, we need to choose two students from the class, one for president and one for vice president.
There are 12 choices for the first position, since any one of the 12 students can be the president. After the president has been chosen, there are 11 students left in the class, so there are 11 choices for the vice president position.
Using the multiplication principle, we can multiply these numbers to get the total number of ways in which a president and vice president can be elected in a class of 12:
12 x 11 = 132
Therefore, there are 132 ways to elect a president and vice president in a class of 12.
A coin is tossed and standard number cube is rolled. What is the probability that the coin lands on heads and the number cube lands on an even number.
The probability of the coin landing on heads is 1/2, since there are two equal and likely outcomes (heads or tails) when a coin is tossed.
The probability of a standard number cube landing on an even number is 3/6, or 1/2, since there are three even numbers (2, 4, and 6) out of six possible outcomes.
To find the probability that both events occur together (i.e. the probability of the intersection of the two events), we need to multiply their probabilities:
P(coin lands on heads and number cube lands on even number) = P(coin lands on heads) x P(number cube lands on even number)
P(coin lands on heads and number cube lands on even number) = 1/2 x 1/2
P(coin lands on heads and number cube lands on even number) = 1/4
Therefore, the probability that the coin lands on heads and the number cube lands on an even number is 1/4.
How many ways can seven students line up for a class picture?
The number of ways to line up seven students for a class picture can be found using the permutation formula:
nPr = n! / (n-r)!
where n is the total number of students and r is the number of students being lined up.
In this case, we have n = 7 and r = 7, since we are lining up all seven students. Therefore, the number of ways to line up seven students is:
7P7 = 7! / (7-7)! = 7! / 0! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040
Therefore, there are 5,040 ways to line up seven students for a class picture.