Exercise 40 Page 705

We want to find out how many groupings there are to capture 5 tetras in an aquarium containing 27 tetras. Note that, in this case, the order is not important. Therefore, the situation involves combinations. To calculate the answer, we will use the formula for combinations of n objects taken r at a time. _n C_r = _n P_r/r! ⇔ _n C_r = n!/(n-r)! r! In our exercise, n=27 and r=5. Let's substitute these values into the above formula.

_n C_r = n!/(n-r)! r!

SubstituteII

n= 27, r= 5

_(27) C_5 = 27!/( 27- 5)! 5!

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Simplify

SubTerm

Subtract term

_(27) C_5 = 27!/22! 5!

Write as a product

_(27) C_5 = 27* 26* 25* 24* 23* 22!/22! 5!

ReduceFrac

a/b=.a /22!./.b /22!.

_(27) C_5 = 27* 26* 25* 24* 23/5!

Write as a product

_(27) C_5 = 27* 26* 25* 24* 23/5* 4* 3* 2* 1

SplitIntoFactors

Split into factors

_(27) C_5 = 27* 26* 5* 5* 4* 3* 2* 23/5* 4* 3* 2* 1

CancelCommonFac

Cancel out common factors

_(27) C_5 = 27* 26* 5* 5* 4* 3* 2* 23/5* 4* 3* 2* 1

SimpQuot

Simplify quotient

_(27) C_5 = 27* 26* 5* 23/1

DivByOne

a/1=a

_(27) C_5 = 27* 26* 5* 23

Multiply

Multiply

_(27) C_5 = 80 730

There are 80 730 groupings to capture 5 tetras in the aquarium containing 27 tetras.