Selections without Order
You already know how to calculate $C_{n,k}$ using the factorial formula. In this section you give it its own name and notation —the binomial coefficient— and use it in more demanding situations: equations, selections with conditions, paths with restrictions, and distributions with a guaranteed minimum.
The binomial coefficient
Find the value of $x$ in the following equality of combinations, knowing that $x \neq 4$: $\binom{x}{4} = \binom{x}{7}$
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: $\binom{8}{3} + \binom{8}{4}$
Calculate the value of the following expression using the properties of combinations:
$\binom{8}{3} + \binom{8}{4}$
Combinatorial equations
When the unknown appears inside a binomial coefficient, the strategy is to expand the factorial formula, simplify, and solve the resulting equation (often linear or quadratic).
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 $n$:
$C_{n, 2} = 45$
Find the value of $x$ that satisfies the following combinatorial equation: $\binom{x}{2} + \binom{x-1}{2} = 25$
Find the value of $x$ that satisfies the following combinatorial equation:
$\binom{x}{2} + \binom{x-1}{2} = 25$
Selections with restrictions
Many combination problems require an additional condition to be met: a fixed number of each "type", or the compulsory inclusion of a particular element. The key is to split the count into blocks and multiply.
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.
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.
Paths on a grid
Binomial coefficients also count shortest paths: if you have to take $m$ steps to the right and $p$ steps up, each path is a way of choosing which $m$ of the $m+p$ positions hold the "right" steps.
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.
Combinations with repetition and minimums
What if, when choosing repeated units from several types, each type must appear at least once? It is solved in two steps: first one compulsory unit of each type is set aside, and then the rest are distributed without restriction.
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.
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…
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.