Amayor has a friend in publishing who tells him they have just published 20 new books and tells him he may pick 10 of them.
a.) How many different selections can the mayor make?
b.)If the govenor picks ten distinct books, how many ways can he make his selection?
a.)one selection
b.)ten ways
a.) To find the number of different selections the mayor can make, we need to use the concept of combinations. Since the mayor can pick any 10 books out of the 20 that were published, we use the formula for combinations, which is given by:
nCr = n! / (r!(n-r)!)
In this case, n = 20 (the total number of books published) and r = 10 (the number of books the mayor may pick). Plugging these values into the formula:
20C10 = 20! / (10!(20-10)!)
Simplifying further:
20C10 = 20! / (10! * 10!)
Now, computing the factorial values:
20! = 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10!
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Plugging these back into the formula:
20C10 = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10!) / (10! * 10!)
We can cancel out the common terms in the numerator and denominator:
20C10 = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / (10!)
Since the numerator does not contain any factorial terms, we can simplify further:
20C10 = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
Now, calculating the final result:
20C10 = 184,756 different selections
Therefore, the mayor can make 184,756 different selections of 10 books from the 20 published.
b.) Since the governor also wants to pick ten distinct books, the number of ways he can make his selection is the same as the number of selections the mayor can make, which we found to be 184,756. Therefore, there are ten different ways the governor can make his selection.