The Multiplication Rule for Counting

Multiplication as a Combinatorial Short Cut

One of the first combinatorial short cuts to counting students learn in school has to do with areas of rectangles. If we have a set of objects to be counted that can be physically arranged into a rectangular shape, then we can use multiplication to do the counting for us. Consider this set of objects ():

Certainly we can count them by pointing and running through the numbers, but it’s more efficient to group them ().

If we group the balls by 4s, we see that we have 6 groups (or, we can see this arrangement as 4 groups of 6 balls). Since multiplication is repeated addition (i.e., \(6\times 4=4+4+4+4+4+4\)), we can use this grouping to quickly see that there are 24 balls.

Let’s generalize this idea a little bit. Let’s say that we’re visiting a bakery that offers customized cupcakes. For the cake, we have three choices: vanilla, chocolate, and strawberry. Each cupcake can be topped with one of four types of frosting: vanilla, chocolate, lemon, and strawberry. How many different cupcake combinations are possible? We can think of laying out all the possibilities in a grid, with cake choices defining the rows and frosting choices defining the columns (Figure 7.5).

Since there are 3 rows (cakes) and 4 columns (frostings), we have \(3\times 4=12\) possible combinations. This is the reasoning behind the Multiplication Rule for Counting, which is also known as the Fundamental Counting Principle. This rule says that if there are \(n\) ways to accomplish one task and \(m\) ways to accomplish a second task, then there are \(n\times m\) ways to accomplish both tasks. We can tack on additional tasks by multiplying the number of ways to accomplish those tasks to our previous product.

Using the Multiplication Rule for Counting

Try it.

Every card in a standard deck of cards has two identifying characteristics: a suit (clubs, diamonds, hearts, or spades; these are indicated by these symbols, respectively: \(♣\), \(♢\), \(♡\), \(♠\)) and a rank (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king; the letters A, J, Q, and K are used to represent the words). Each possible pair of suit and rank appears exactly once in the deck. How many cards are in the standard deck?

Solution

Since there are 4 suits and 13 ranks, the number of cards must be \(4\times 13=52\) ().

Condensed: the full section is in OpenStax Contemporary Mathematics.

Haddaad Ma jiro xisaabis oo ku nool midkan, laakiin qaybaha ka mid ah waa la xisaabi karaa. Ku raaxayso mid ka hooseeya, ama ku qor adiga kuu gaar ah.

⌨ Nooc ✎ Sawir

Su'aalkaaga xisaabta

π Xalin

Tababar (3)

Ku day mid kasta oo ka mid ah oo ku saabsan buugaag hore. Muuji jawaabta si aad u hubiso; kuwa la xaqiijiyay waxaa la furi karaa in xalka ah tallaabo kasta.

1. Every card in a standard deck of cards has two identifying characteristics: a suit (clubs, diamonds, hearts, or spades; these are indicated by these symbols, respectively: \(♣\), \(♢\), \(♡\), \(♠\)) and a rank (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king; the letters A, J, Q, and K are used to represent the words). Each possible pair of suit and rank appears exactly once in the deck. How many cards are in the standard deck?

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   Since there are 4 suits and 13 ranks, the number of cards must be \(4\times 13=52\) ().

2. The University Combinatorics Club has 31 members: 8 seniors, 7 juniors, 5 sophomores, and 11 first-years. How many possible 4-person committees can be formed by selecting 1 member from each class?

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   Since we have 8 choices for the senior, 7 choices for the junior, 5 for the sophomore, and 11 for the first-year, there are \(8\times 7\times 5\times 11=3,080\) different ways to fill out the committee.

3. The standard license plates for vehicles in a certain state consist of 6 characters: 3 letters followed by 3 digits. There are 26 letters in the alphabet and 10 digits (0 through 9) to choose from. How many license plates can be made using this format?

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   Since there are 26 different letters and 10 different digits, the total number of possible license plates is \(26\times 26\times 26\times 10\times 10\times 10=17,576,000\).

Ku hay shaqadaada

A xisaab bilaash ah oo ku daray qoraalada waxbarasho kasta, diiwaanka waxa aad dhamaatay, dhibaatooyinka aad xal u tahay meel kaliya, iyo macallinka aad weydiisan kartaa oo ku saabsan bogga. Xisaabta oo keliya waa u furan yahay qof kasta, la diiwaan geliyey ama aan.

Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

iimaginary unit
i² = −1.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\approxapproximately equal
Equal to the precision shown, not exactly.n!factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.\binom{n}{k}binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).\sum_{k=1}^{n} a_ksummation
Add a_k for k = 1 up to n.A \cup Bunion, intersection, difference
In either; in both; in A but not B.\bar{x}sample mean, population mean
Average of the data; average of the whole population.\sigmastandard deviation, sample s.d., variance
Typical distance from the mean; its square.P(A)probability, conditional probability
Chance of A; chance of A given that B happened.E[X]expected value, variance
Probability-weighted average of X; its spread.N(\munormal distribution, z-score
The bell curve with mean μ and variance σ²; (x − μ)/σ.

Sida loo: The Multiplication Rule for Counting

1. Apply the Multiplication Rule for Counting to solve problems.
2. combinatorics
3. Multiplication Rule for Counting (Fundamental Counting Principle)
4. The Multiplication Rule for Counting is used to count large sets.

Su'aalaha dadku waydiiyaan

Mean or median, which should I use?

Median when the data have outliers or a long tail (incomes, house prices); mean when the data are roughly symmetric and you want every value to count. Report both if they disagree. The gap is itself information.

What does a p-value actually say?

The probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability that the null hypothesis is true.

Why divide by n − 1 for the sample variance?

The sample mean sits closer to the sample than the true mean does, so squared deviations from it are slightly too small on average; dividing by n − 1 instead of n corrects the bias.

Qaybaha ee boggaan waxaa la beddelay ka OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed iyo dib-oo halkan lagu sharaxay; khaladaadka waa inoo. Sababtoo ah asalka waxaa loo oggolaaday in aan loo isticmaalin ganacsi, fasalkan waa bilaash, waxba ma iibiyo, iyo waxaa lagu wadaagayaa liisan la mid ah.

In ka badan Statistics & Probability

Sampling and dataDescribing data with graphsMean, median and modeProbabilityCounting: permutations and combinationsDiscrete random variablesContinuous random variablesThe normal distributionThe central limit theoremConfidence intervalsHypothesis testingComparing two samplesChi-square testsLinear regression and correlation