Exercises: Unordered

$14$ players are taking part in a chess tournament. If each player must face everyone else exactly once, determine the total number of games that will be played in the tournament.

To prepare a salad, you have $8$ different ingredients available (lettuce, tomato, carrot, onion, cucumber, tuna, corn, and olives). Calculate how many different salads of $4$ ingredients you can make.

An exam consists of $10$ questions. The teacher indicates that students must choose and answer only $6$ of them. Determine in how many different ways a student can select the questions to answer.

There are $12$ people in a business meeting. At the end, everyone says goodbye by shaking hands exactly once. Calculate how many handshakes take place in total.

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

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

Calculate the value of the following expression using the properties of binomial coefficients:

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

$3$ cards are drawn simultaneously at random from a $40$-card Spanish deck. Determine how many different results can be obtained.

$7$ points are marked on a circle. Calculate how many different line segments can be drawn by joining two of these points.

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 the staff, calculate how many different teams can be formed.

A bakery sells $6$ different types of cakes. Determine in how many different ways $4$ cakes can be bought, keeping in mind that several of the same type can be chosen.

Calculate the total number of diagonals that can be drawn in a regular polygon of $10$ sides (decagon).

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

$C_{n, 2} = 45$

$5$ cards are drawn simultaneously from a $40$-card Spanish deck (divided into $4$ suits of $10$ cards each). Determine in how many of the possible results there are exactly $3$ cards of the coins suit.

A robot moves along a grid from point $A$ to point $B$. It can only move to the right or upward, advancing one intersection at each step. Calculate the number of different paths it can follow.

You have $8$ identical tickets for a concert and you want to divide them among your $4$ best friends. Calculate how many ways you can distribute them, knowing that a friend can receive several tickets or even none.

On a test with $12$ questions, you must answer a total of $8$. However, the first $3$ questions are required. Determine the number of different ways you can complete your selection of questions.

A hospital must 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 the team to contain at least $3$ doctors.

On a grid formed by $9$ points distributed in $3$ rows and $3$ columns, determine how many different triangles can be formed using these points as vertices. Keep in mind that three collinear points do not form a triangle.

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

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

A lottery game consists of choosing $6$ numbers from a total of $49$. If the winning combination of $6$ numbers is already known, calculate how many different possible bets would obtain exactly $4$ correct numbers.

A robot moves along a grid from node $A$ to node $B$, moving only to the right and upward. It must pass through node $C$ to pick up a package. Determine how many valid routes exist.

An ice cream parlor offers $5$ different flavors. You want to buy $8$ tubs of ice cream for a party, but you want to make sure you get at least one tub of each of the $5$ flavors. Calculate how many different ways you can place your order.

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.

The traditional game of dominoes goes up to double-six (the values range from $0$ to $6$). A manufacturer decides to create a "super dominoes" set that goes up to double-nine (the values range from $0$ to $9$). Calculate how many total tiles this new set will have and how many of them will not be doubles.