"Without Order: Combinations and the Binomial Coefficient"
Before you start, make sure you know: Quadratic equations (Secondary Year 3)
In Melate (a Mexican lottery) you choose $6$ numbers out of $56$, and it doesn't matter in what order you mark them. How many different combinations can you play? What is your probability of winning the jackpot? You will calculate it at the end of the page.
Choosing when order doesn't matter
When you form a team, choose ingredients, or select questions for an exam, order doesn't count: the team of Ximena and Diego is the same as the team of Diego and Ximena. These selections are called combinations.
Why do we divide by $r!$? If order mattered, there would be ${}_nP_r$ arrangements. But each group of $r$ elements has been counted $r!$ times, once for each way of ordering it. Dividing by $r!$ leaves each group counted only once.
Some exercises mention a deck of $40$ cards divided into $4$ suits of $10$ cards each. All that matters is that there are $40$ cards in total and $10$ in each suit. Drawing cards "simultaneously" means that order doesn't matter. In the examples on this page we use the standard deck of $52$ cards (4 suits of 13).
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To make a salad, you have $8$ different ingredients available (lettuce, tomato, carrot, onion, cucumber, tuna, sweetcorn and olives). Calculate how many different salads…
To make a salad, you have $8$ different ingredients available (lettuce, tomato, carrot, onion, cucumber, tuna, sweetcorn and olives). Calculate how many different salads of $4$ ingredients you can make.
An exam consists of $10$ questions. The teacher specifies that students must choose and answer only $6$ of them. Determine in how many different ways a student can…
An exam consists of $10$ questions. The teacher specifies that students must choose and answer only $6$ of them. Determine in how many different ways a student can select the questions to answer.
Many everyday situations are really "choose $2$ of $n$": each pair counts only once because a greeting between Regina and Mateo is the same as one between Mateo and Regina.
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In a chess tournament, $14$ players participate. If each player must play against all the others exactly once, determine the total number of matches that will be played…
In a chess tournament, $14$ players participate. If each player must play against all the others exactly once, determine the total number of matches that will be played in the tournament.
In a business meeting, there are $12$ people. At the end, everyone says goodbye by shaking hands once with each other. Calculate how many handshakes take place in total.
In a business meeting, there are $12$ people. At the end, everyone says goodbye by shaking hands once with each other. Calculate how many handshakes take place in total.
Sometimes the group must have fixed numbers of each type, or must include someone without fail. The key is to split the count into blocks and multiply.
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A company needs to form a project team consisting of $3$ engineers and $2$ economists. If there are $8$ engineers and $5$ economists available on staff, calculate how…
A company needs to form a project team consisting of $3$ engineers and $2$ economists. If there are $8$ engineers and $5$ economists available on staff, calculate how many different teams can be formed.
In a questionnaire with $12$ questions, you must answer a total of $8$ questions. However, the first $3$ questions are compulsory. Determine in how many different ways…
In a questionnaire with $12$ questions, you must answer a total of $8$ questions. However, the first $3$ questions are compulsory. Determine in how many different ways you can complete your choice of questions.
A hospital needs to form a night-shift team of $5$ people, selected from a group of $7$ doctors and $6$ nurses. Calculate how many different teams can be formed if…
A hospital needs to form a night-shift team of $5$ people, selected from a group of $7$ doctors and $6$ nurses. Calculate how many different teams can be formed if regulations require that the team contains at least $3$ doctors.
and both the favorable and the possible outcomes are counted with combinations.
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$5$ cards are drawn simultaneously from a Spanish deck of $40$ cards (divided into $4$ suits of $10$ cards each). Determine in how many of the possible outcomes there…
$5$ cards are drawn simultaneously from a Spanish deck of $40$ cards (divided into $4$ suits of $10$ cards each). Determine in how many of the possible outcomes there are exactly $3$ cards of the suit of coins.
The Primitiva lottery game consists of choosing $6$ numbers out of a total of $49$. Given the winning combination of $6$ numbers, calculate how many of all possible…
The Primitiva lottery game consists of choosing $6$ numbers out of a total of $49$. Given the winning combination of $6$ numbers, calculate how many of all possible different bets would get exactly $4$ correct numbers.
In a grid consisting of $9$ points distributed in $3$ rows and $3$ columns, determine how many different triangles can be formed using these points as vertices. Bear in…
In a grid consisting of $9$ points distributed in $3$ rows and $3$ columns, determine how many different triangles can be formed using these points as vertices. Bear in mind that three collinear points do not form a triangle.
An urn contains $6$ red balls, $5$ blue balls and $4$ green balls. $4$ balls are drawn simultaneously. Calculate how many different outcomes contain exactly $2$ red…
An urn contains $6$ red balls, $5$ blue balls and $4$ green balls. $4$ balls are drawn simultaneously. Calculate how many different outcomes contain exactly $2$ red balls and at least $1$ blue ball.
When the unknown is inside a binomial coefficient, expand the formula, simplify and solve. Often you get a quadratic equation, which you already know how to solve by factoring or with the quadratic formula.
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Solve the following equation to find the value of the natural number $n$: $C_{n, 2} = 45$
Solve the following equation to find the value of the natural number $n$:
A robot (or you, on the streets of a neighborhood) goes from $A$ to $B$ moving only right or up. If you have to take $m$ steps to the right and $p$ steps up, each path is a sequence of $m + p$ steps.
To solve the exercises, count in the figure how many steps to the right and how many steps up separate each pair of points.
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A robot moves along a grid from point $A$ to point $B$. It can only move right or up, advancing one intersection per step. Calculate the number of different paths it can…
A robot moves along a grid from point $A$ to point $B$. It can only move right or up, advancing one intersection per step. Calculate the number of different paths it can follow.
A robot moves along a grid from node $A$ to node $B$, moving only to the right and upwards. It must pass through node $C$ to collect a package. Determine how many valid…
A robot moves along a grid from node $A$ to node $B$, moving only to the right and upwards. It must pass through node $C$ to collect a package. Determine how many valid routes exist.
What if you can choose the same type several times? For example, buying $4$ pastries from $6$ types, with several of the same kind. Here only how many of each type you take matters, not the order.
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A bakery sells $6$ different types of cakes. Determine in how many different ways $4$ cakes can be bought, keeping in mind that several cakes of the same type can be…
A bakery sells $6$ different types of cakes. Determine in how many different ways $4$ cakes can be bought, keeping in mind that several cakes of the same type can be chosen.
You have $8$ identical concert tickets and you want to distribute them among your $4$ best friends. Calculate in how many ways you can distribute them, knowing that a…
You have $8$ identical concert tickets and you want to distribute them among your $4$ best friends. Calculate in how many ways you can distribute them, knowing that a friend may receive several tickets or even none.
The traditional game of dominoes goes up to double six (the scores range from $0$ to $6$). A manufacturer decides to create a "super dominoes" set that goes up to double…
The traditional game of dominoes goes up to double six (the scores range from $0$ to $6$). A manufacturer decides to create a "super dominoes" set that goes up to double nine (the scores range from $0$ to $9$). Calculate how many tiles in total this new set will have and how many of them will not be doubles.
If each type must appear at least once, first set aside one piece of each type and distribute the rest freely.
Back to the beginning: in Melate you choose $6$ numbers out of $56$ without regard to order, so there are $\binom{56}{6} = 32{,}468{,}436$ possible combinations. Only one wins the jackpot, so
An ice cream shop offers $5$ different flavors. You want to buy $8$ tubs of ice cream for a party, but you want to make sure you take at least one tub of each of the $5$…
An ice cream shop offers $5$ different flavors. You want to buy $8$ tubs of ice cream for a party, but you want to make sure you take at least one tub of each of the $5$ flavors. Calculate in how many different ways you can place your order.