Exercise 26 Page 842

We want to find the probability of choosing a specific set of 7 books off a bookshelf holding 12 books. To do so, we will use the theoretical probability. P=Favorable Outcomes/Possible Outcomes We start by finding the number of total possible outcomes. This will be the number of combinations in which we can choose those particular 7 books from a shelf of 12 books. The order in which we choose them is not important since we are only interested in a set of books, not their order. Let's recall the formula for the number of combinations of n objects taken r at a time. _nC_r = n!/r!(n-r)! In our case there are 12 books on the shelf, so n= 12. Out of them, 7 books are randomly selected, so r = 7. Let's substitute these values and find the number of possible combinations.

_nC_r = n!/(n-r)! * r!

SubstituteII

n= 12, r= 7

_(12) C_2 = 12!/7!( 12- 7)!

▼

Evaluate right-hand side

SubTerm

Subtract term

_(12)C_2 = 12!/7! 5!

Write as a product

_(12)C_2 = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1/( 7 * 6 * 5 * 4 * 3 * 2 * 1 ) ( 5 * 4 * 3 * 2 * 1 )

CancelCommonFac

Cancel out common factors

_(12)C_2 = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1/( 7 * 6 * 5 * 4 * 3 * 2 * 1 ) ( 5 * 4 * 3 * 2 * 1 )

SimpQuot

Simplify quotient

_(12)C_2 = 12 * 11 * 10 * 9 * 8/5 * 4 * 3 * 2 * 1

Multiply

Multiply

_(12)C_2 = 95 040/120

CalcQuot

Calculate quotient

_(12)C_2 = 792

The total number of possible outcomes is 792. Next, we will look for the number of favorable outcomes. Note that there is only one combination that chooses a specific fixed set of 7 books, so there is only one favorable outcome. We have enough information to calculate the desired probability. P=Favorable Outcomes/Possible Outcomes [0.9em] ⇕ P=1/792