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C ( 5 , 0 ) + C ( 5 , 1 ) + C ( 5 , 2 ) + C ( 5 , 3 ) + C ( 5 , 4 ) + C ( 5 , 5 ) = 32

There are 32 possible pizzas. This result is equal to 2 5 .

We are presented with a sequence of choices. For each of the n objects we have two choices: include it in the subset or not. So for the whole subset we have made n choices, each with two options. So there are a total of 2 · 2 · 2 · · 2 possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time.

Formula for the number of subsets of a set

A set containing n distinct objects has 2 n subsets.

Finding the number of subsets of a set

A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?

We are looking for the number of subsets of a set with 4 objects. Substitute n = 4 into the formula.

2 n = 2 4      = 16

There are 16 possible ways to order a potato.

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A sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?

64 sundaes

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Finding the number of permutations of n Non-distinct objects

We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be 12 ! ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the 12 ! permutations we counted are duplicates. The general formula for this situation is as follows.

n ! r 1 ! r 2 ! r k !

In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are 4 ! ways to order the stars and 3 ! ways to order the moon.

12 ! 4 ! 3 ! = 3 , 326 , 400

There are 3,326,400 ways to order the sheet of stickers.

Formula for finding the number of permutations of n Non-distinct objects

If there are n elements in a set and r 1 are alike, r 2 are alike, r 3 are alike, and so on through r k , the number of permutations can be found by

n ! r 1 ! r 2 ! r k !

Finding the number of permutations of n Non-distinct objects

Find the number of rearrangements of the letters in the word DISTINCT.

There are 8 letters. Both I and T are repeated 2 times. Substitute n = 8 ,   r 1 = 2 ,   and   r 2 = 2   into the formula.

8 ! 2 ! 2 ! = 10 , 080  

There are 10,080 arrangements.

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Find the number of rearrangements of the letters in the word CARRIER.

840

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Access these online resources for additional instruction and practice with combinations and permutations.

Key equations

number of permutations of n distinct objects taken r at a time P ( n , r ) = n ! ( n r ) !
number of combinations of n distinct objects taken r at a time C ( n , r ) = n ! r ! ( n r ) !
number of permutations of n non-distinct objects n ! r 1 ! r 2 ! r k !

Key concepts

  • If one event can occur in m ways and a second event with no common outcomes can occur in n ways, then the first or second event can occur in m + n ways. See [link] .
  • If one event can occur in m ways and a second event can occur in n ways after the first event has occurred, then the two events can occur in m × n ways. See [link] .
  • A permutation is an ordering of n objects.
  • If we have a set of n objects and we want to choose r objects from the set in order, we write P ( n , r ) .
  • Permutation problems can be solved using the Multiplication Principle or the formula for P ( n , r ) . See [link] and [link] .
  • A selection of objects where the order does not matter is a combination.
  • Given n distinct objects, the number of ways to select r objects from the set is C ( n , r ) and can be found using a formula. See [link] .
  • A set containing n distinct objects has 2 n subsets. See [link] .
  • For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See [link] .

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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