Combinatorics
Ximena forgot the 4-digit code on her bicycle lock. If she tried them one by one, how many codes would she have to check at most? Counting without writing out every possibility is exactly what combinatorics does: you identify which type of count it is and apply its formula.
The multiplication principle and variations with repetition
If a choice is made in several stages, and stage 1 has $n_1$ options, stage 2 has $n_2$, and so on, the total number of outcomes is the product $n_1 \times n_2 \times \cdots$. This is the multiplication principle: each option from one stage is combined with all the options from the next.
A restaurant offers $4$ starters, $5$ main courses and $3$ desserts on its set menu. Calculate how many different complete meals (consisting of a starter, a main course…
A restaurant offers $4$ starters, $5$ main courses and $3$ desserts on its set menu. Calculate how many different complete meals (consisting of a starter, a main course and a dessert) a customer can order.
A digital bicycle lock is opened using a $4$-digit code (from $0$ to $9$). Determine how many different codes can be set if repeating digits is allowed.
A digital bicycle lock is opened using a $4$-digit code (from $0$ to $9$). Determine how many different codes can be set if repeating digits is allowed.
Variations without repetition: choosing and ordering
Now each element that has been used is no longer available. If you choose $k$ of $n$ elements in order, the first position has $n$ options, the second has $n-1$, and so on down to $n-k+1$.
In equations with $V_{n,k}$, it helps to write each variation as a product of factors and cancel the common factors.
A robotics club has $12$ members. Determine how many different ways a committee consisting of a president, a vice president and a secretary can be chosen, given that no…
A robotics club has $12$ members. Determine how many different ways a committee consisting of a president, a vice president and a secretary can be chosen, given that no person can hold more than one position.
A security system requires a password consisting of $2$ distinct vowels followed by $3$ distinct digits (from $0$ to $9$). Calculate how many different passwords can be…
A security system requires a password consisting of $2$ distinct vowels followed by $3$ distinct digits (from $0$ to $9$). Calculate how many different passwords can be configured under these conditions.
Solve the following combinatorial equation to find the natural value of $n$: $3 \times V_{n, 3} = 4 \times V_{n-1, 3}$
Solve the following combinatorial equation to find the natural value of $n$:
$3 \times V_{n, 3} = 4 \times V_{n-1, 3}$
Permutations: arranging all the elements
Arranging all $n$ elements of a set at once is a case of variation with $k=n$: $P_n = n!$.
If some elements are repeated ($n_1$ of one kind, $n_2$ of another, …), swapping two identical ones does not produce a new arrangement, so you divide by the internal arrangements of each kind.
Calculate how many different ways $6$ different textbooks can be arranged in a line on a shelf.
Calculate how many different ways $6$ different textbooks can be arranged in a line on a shelf.
Calculate the number of different anagrams, whether meaningful or not, that can be formed using all the letters of the word $INGENIERIA$.
Calculate the number of different anagrams, whether meaningful or not, that can be formed using all the letters of the word $INGENIERIA$.
The letters of the word $AEROPUERTO$ are shuffled at random to form a new anagram. Calculate the probability that the new word starts and ends with the letter $O$.
The letters of the word $AEROPUERTO$ are shuffled at random to form a new anagram. Calculate the probability that the new word starts and ends with the letter $O$.
Arranging with conditions: elements kept together and round tables
Elements that must stay together. Treat them as a single block, arrange the blocks, and then multiply by the number of ways to arrange the elements inside the block.
Round tables. If all that matters is who sits next to whom (rotating everyone does not change the arrangement), you fix one person and arrange the others: $n$ people can be seated in $(n-1)!$ ways.
Seven friends go to the cinema and sit in a row of $7$ seats. Two of them, Ana and Bruno, must sit next to each other. Calculate how many different ways they can occupy…
Seven friends go to the cinema and sit in a row of $7$ seats. Two of them, Ana and Bruno, must sit next to each other. Calculate how many different ways they can occupy the seats.
Four men and four women are sitting down to dinner around a circular table. Determine how many different ways they can be seated if people of the same gender cannot sit…
Four men and four women are sitting down to dinner around a circular table. Determine how many different ways they can be seated if people of the same gender cannot sit next to each other (that is, men and women must alternate). Consider two arrangements to be identical if one can be obtained by rotating the other.
Combinations: choosing without order
When all that matters is which elements are chosen and not the order, you use combinations. Each group of $k$ elements appears $k!$ times among the variations, which is why you divide by $k!$.
A technology teacher needs to select $4$ students from a class of $20$ to attend a science fair. Calculate how many different groups of representatives can be formed.
A technology teacher needs to select $4$ students from a class of $20$ to attend a science fair. Calculate how many different groups of representatives can be formed.
$8$ points are marked on a plane, no three of which are collinear. Determine how many different straight lines can be drawn by connecting pairs of these points.
$8$ points are marked on a plane, no three of which are collinear. Determine how many different straight lines can be drawn by connecting pairs of these points.
In a chess tournament where each participant plays exactly one match against every other player, a total of $105$ matches were played. Determine the number of players…
In a chess tournament where each participant plays exactly one match against every other player, a total of $105$ matches were played. Determine the number of players who participated in the tournament.
Several groups, "at least," and distributions
In real problems, techniques are combined. Choosing $k$ from one group and $j$ from another: you multiply the combinations. Alternative cases that cannot happen at the same time: you add.
Distributing people into named groups. You choose the members of the first group, then those of the second from the people who remain, and so on.
A biomedical research team must consist of $5$ people. The available candidates are $6$ biologists and $4$ chemists. Determine how many different teams can be formed if…
A biomedical research team must consist of $5$ people. The available candidates are $6$ biologists and $4$ chemists. Determine how many different teams can be formed if the team must contain exactly $3$ biologists and $2$ chemists.
A tech company is going to form a cybersecurity committee of $6$ experts chosen from $8$ software engineers and $7$ network analysts. Calculate how many different…
A tech company is going to form a cybersecurity committee of $6$ experts chosen from $8$ software engineers and $7$ network analysts. Calculate how many different committees can be formed if there must be at least $4$ software engineers in the group.
The European Space Agency needs to assign $12$ astronauts to $3$ different space modules (Command Module, Lunar Module and Service Module). Calculate how many different…
The European Space Agency needs to assign $12$ astronauts to $3$ different space modules (Command Module, Lunar Module and Service Module). Calculate how many different ways the crew members can be distributed if each module must accommodate exactly $4$ astronauts.
Grid paths and distributions with zeros
Shortest paths. On a grid, to go from $(0,0)$ to $(a,b)$ you can only move right ($R$) or up ($U$), with $a$ steps $R$ and $b$ steps $U$ in total. Each path is a way of arranging those letters: $C_{a+b,\,a}$.
Distributing identical objects (zeros allowed). Distributing $m$ identical objects into $r$ boxes is equivalent to arranging $m$ objects and $r-1$ dividers: $C_{m+r-1,\;r-1}$.
In a city whose streets form a perfect grid, calculate how many paths of minimum length exist to go from point $A(0,0)$ to point $B(6,5)$, passing through intersection…
In a city whose streets form a perfect grid, calculate how many paths of minimum length exist to go from point $A(0,0)$ to point $B(6,5)$, passing through intersection $C(3,2)$.
An investor wishes to divide $200\,000$ pesos among $4$ startups. If the investment in each company must be an exact multiple of $20\,000$ pesos (including the…
An investor wishes to divide $200\,000$ pesos among $4$ startups. If the investment in each company must be an exact multiple of $20\,000$ pesos (including the possibility of investing $0$ pesos in any of them), determine how many different ways the capital can be distributed.
Combinatorics and probability
With equally likely events, Laplace's rule says $P(A) = \dfrac{\text{favorable outcomes}}{\text{possible outcomes}}$. Combinatorics is used to count both.
If you draw without replacement and order does not matter, you count groups with combinations: possible outcomes $C_{N,n}$ and favorable outcomes, the product of the combinations of each type.
From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. Calculate the probability that all three cards are Kings.
From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. Calculate the probability that all three cards are Kings.
A batch of $20$ microchips contains $4$ defective ones. If $3$ microchips are selected at random without replacement for inspection, calculate the probability that…
A batch of $20$ microchips contains $4$ defective ones. If $3$ microchips are selected at random without replacement for inspection, calculate the probability that exactly $1$ of them is defective.
From a standard deck of $52$ cards, a hand of $5$ cards is dealt. Calculate the probability that the player gets exactly $3$ Aces in their initial hand.
From a standard deck of $52$ cards, a hand of $5$ cards is dealt. Calculate the probability that the player gets exactly $3$ Aces in their initial hand.
Probability with several types of elements
When there are several types (colors, card values, good and bad parts), the favorable outcomes are built by choosing how many of each type and multiplying. If the event includes several cases ("at most," "at least"), you add them.
Back to the beginning: Ximena's lock has $10^4 = 10{,}000$ possible codes. If she tries one at random, the probability of getting it right on the first try is $\dfrac{1}{10{,}000}$.
An urn contains $5$ red balls, $4$ blue balls and $3$ green balls. If $3$ balls are drawn simultaneously at random, calculate the probability that they are all of…
An urn contains $5$ red balls, $4$ blue balls and $3$ green balls. If $3$ balls are drawn simultaneously at random, calculate the probability that they are all of different colors.
In a quality control process, a random sample of $5$ items is taken from a production line containing a total of $50$ items, $10\%$ of which are defective. Calculate the…
In a quality control process, a random sample of $5$ items is taken from a production line containing a total of $50$ items, $10\%$ of which are defective. Calculate the probability of finding at most $1$ defective item in the sample taken.
$5$ cards are drawn at random from a standard deck of $52$ cards. Calculate the probability of obtaining a full house, that is, a hand consisting of $3$ cards of one…
$5$ cards are drawn at random from a standard deck of $52$ cards. Calculate the probability of obtaining a full house, that is, a hand consisting of $3$ cards of one rank and $2$ cards of another rank.