Selections without Order

Find the value of $x$ in the following equality of combinations, knowing that $x \neq 4$:

$\binom{x}{4} = \binom{x}{7}$

Calculate the value of the following expression using the properties of combinations:

$\binom{8}{3} + \binom{8}{4}$

In a 4th Year class of 24 students, a student council delegation of 2 representatives must be chosen to attend the school's meetings.

1. a) Before applying any formula, make a reasoned estimate without doing meticulous calculations: do you think the number of possible pairs will be in the tens, hundreds, or thousands? Support your answer by thinking about the options each student has.
2. b) Model the situation using the appropriate combination formula, carry out the calculation, and check whether your initial estimate was accurate.
3. c) Based on the symmetry property of combinations $\binom{n}{k} = \binom{n}{n-k}$, explain why choosing the 2 representatives of the delegation is mathematically equivalent to choosing the 22 students who are left out, without needing to calculate additional factorials.

Solve the following equation to find the value of the natural number $n$:

$C_{n, 2} = 45$

Find the value of $x$ that satisfies the following combinatorial equation:

$\binom{x}{2} + \binom{x-1}{2} = 25$

In a regional federated padel tournament in Castilla-La Mancha, the preliminary stage is played in independent groups. In each group, all registered pairs play exactly one match against every other pair in their group (single round-robin system). When closing the preliminary stage scoresheet, the organizers confirm that exactly 45 matches were played in total in Group A.

1. a) Write a combinatorial equation using the notation $\binom{n}{k}$ involving the unknown number $n$ of registered pairs in Group A.
2. b) Algebraically expand the combination to transform the equality into a quadratic equation in standard form.
3. c) Solve the equation, analyze the two solutions obtained, reasonably discarding the one that makes no sense in this context, and state how many pairs were competing in that group.

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.

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.

$5$ cards are drawn simultaneously from a 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 diamonds.

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 routes exist.

In the orthogonal grid of streets in Georgian Dublin (around Merrion Square), a bicycle courier needs to travel from intersection A to intersection B. To optimize their route, the courier only moves east or north at each intersection, requiring an exact path of 5 blocks east and 3 blocks north (a total of 8 blocks of travel).

1. a) Model the path using combinatorics: explain why any optimal route of 8 blocks is equivalent to choosing the steps taken in one of the two directions, and calculate how many different routes connect A to B using the expression $\binom{m+p}{m}$.
2. b) The Gardaí have closed the intermediate intersection C for construction, located exactly 2 blocks east and 1 block north of point A. Model how many of the possible routes would necessarily pass through intersection C and deduce how many alternative construction-free routes remain available to the courier.
3. c) Suppose the courier must deliver an urgent package at intersection D (located 4 blocks east and 2 north of A) and that intersection C remains closed. Set up and calculate the combinatorial expression determining how many different routes allow going from A to B necessarily passing through D while avoiding the closed intersection C at all times.

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 friend may receive several tickets or even none.

An ice cream shop offers $5$ different flavours. 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 $5$ flavours. Calculate in how many different ways you can place your order.