Permutation
For a segment of a radio, a disc jockey can play 7 songs. If there are 13 songs to select from, in how many ways can the program for this segment be arranged?
Question # 2
A corporation has ten members on its board of directors. In how many different ways can it elect a president, vice president, secretary, and treasurer?
To find the number of ways to arrange songs for the radio segment, we need to use the concept of permutations. A permutation is an arrangement of objects in a specific order.
In this case, we have 13 songs to select from, and we need to choose 7 songs to play. The order in which the songs are played matters. Therefore, we can use the formula for permutations:
nPr = n! / (n - r)!
Where n is the total number of objects (songs) and r is the number of objects to be chosen (7 songs). "!" denotes factorial, which is the product of all positive integers less than or equal to a given number.
Using the formula, we can calculate the number of ways to arrange the songs:
P(13, 7) = 13! / (13 - 7)!
= 13! / 6!
= (13 x 12 x 11 x 10 x 9 x 8 x 7 x 6!) / 6!
= 13 x 12 x 11 x 10 x 9 x 8
= 62,635,680
Therefore, there are 62,635,680 different ways to arrange the songs for the radio segment.
Now, let's move on to Question #2.
To find the number of different ways to elect a president, vice president, secretary, and treasurer for the board of directors, we can again use permutations.
We have 10 members on the board, and we need to choose 4 members for the positions. The order in which they are selected matters.
Using the permutation formula:
nPr = n! / (n - r)!
Where n represents the total number of objects (10 members) and r represents the number of objects to be chosen (4 positions).
P(10, 4) = 10! / (10 - 4)!
= 10! / 6!
= (10 x 9 x 8 x 7 x 6!) / 6!
= 10 x 9 x 8 x 7
= 5,040
Therefore, there are 5,040 different ways to elect a president, vice president, secretary, and treasurer for the board of directors.