Exercises: Factorial

Calculate the exact value of the expression $6! - 4!$.

Simplify the following fraction of factorials without using a calculator: $\frac{8!}{6!}$.

Express the numerical product $12 \cdot 11 \cdot 10$ using a division of factorials.

Calculate the result of the following quotient: $\frac{7!}{5! \cdot 2!}$.

Express the algebraic product $n \cdot (n-1) \cdot (n-2)$ as a quotient of factorials.

Simplify the algebraic expression $\frac{(n+1)!}{n!}$, giving the result in terms of $n$.

Work out the value of $\frac{100!}{99!}$ mentally.

Determine the value of the natural number $x$ given that $x! = 120$.

In a 4th-year ESO class, the teacher asks to simplify the fraction $\dfrac{8!}{4!}$. Sergio performs the following operation on the board:

1. a) Explain in your own words why Sergio's reasoning is incorrect, relying on the meaning of the factorial as a product of consecutive integers.
2. b) Correctly expand the expression $\dfrac{8!}{4!}$ by simplifying the common factors before multiplying, and find its true value.

In the final gala of the Benidorm Fest festival broadcast by RTVE, 6 qualified artists must perform on stage to compete for a spot in Eurovision.

1. a) Model this situation: express using a factorial in how many different ways the production team can order the performances of the 6 participants throughout the night, and calculate this total number.
2. b) If the organisation decides by prior draw to fix which artist will perform first and which artist will close the gala in sixth place, in how many ways can the other 4 artists now be ordered? Justify the result by writing the appropriate operation with factorials.

Simplify the algebraic expression $\frac{n!}{(n-2)!}$ as far as possible, expanding the numerator until terms can be cancelled with the denominator.

Solve the equation $\frac{(x+2)!}{(x+1)!} = 15$, stating the value of $x$.

In a $100$ metres sprint, $8$ athletes take part. Using a factorial, express in how many different ways they can cross the finish line, assuming there are no ties, and calculate its final value.

Express the polynomial expression $(x+3)(x+2)(x+1)$ as a quotient of factorials, assuming that $x$ is a natural number.

Simplify the expression $\frac{(n+1)! - n!}{n!}$ by taking out a common factor in the numerator.

A security system requires a code consisting of $5$ distinct letters chosen between $A, B, C, D$ and $E$. Use factorials to determine and calculate how many different codes can be created.

Check whether the equality $4! + 3! = 7!$ is true or false, justifying your answer by calculating the corresponding factorials.

Solve the equation $\frac{x!}{(x-1)!} = x^2 - 12$, finding the possible value or values for the natural number $x$.

A team of Renfe Cercanías inspectors in Madrid must check 5 consecutive stations chosen from a line that has a total of 8 stations ($A, B, C, D, E, F, G, H$), where the exact order in which they are visited matters. The number of possible ordered routes is given by the expression $\dfrac{8!}{(8-5)!}$.

1. a) Without performing complete calculations on paper or using a calculator, make an estimate: do you think there will be more or fewer than 5000 possible routes? Justify your conjecture by mentally approximating the product of the numbers that will remain unsimplified.
2. b) Expand the expression by simplifying the factorial in the denominator and obtain the exact number of possible itineraries.
3. c) Check whether your initial estimate was accurate by calculating the difference between the actual value and the estimated 5000, and explain why this type of calculation grows so rapidly as the number of elements increases.

To organise the night on-call shifts at a local Cruz Roja post, it is necessary to calculate in how many ways two posts of different responsibility (ambulance driver and emergency technician) can be assigned among the 9 available volunteers. The applied formula leads to the calculation of $\dfrac{9!}{7! \cdot 1!}$.

Two team members propose different strategies:

• Pablo's strategy: First calculate $9! = 362\,880$ and $7! = 5\,040$, and then perform the division $\dfrac{362\,880}{5\,040 \cdot 1}$.

• Lucía's strategy: Express $9!$ as $9 \cdot 8 \cdot 7!$, directly cancel the term $7!$ present in both the numerator and the denominator, and solve $9 \cdot 8$.

1. a) Apply Lucía's strategy to obtain the number of ways to assign the shift.
2. b) Compare both procedures and write an argument justifying which of the two is more convenient when performing calculations by hand or with larger numbers (for example, with $\dfrac{100!}{98!}$).

Solve the quadratic equation resulting from the expression $\frac{n!}{(n-2)!} = 42$, stating the only valid value for the natural number $n$.

Prove algebraically that $n! + (n-1)! = (n+1) \cdot (n-1)!$ for any natural number $n \geq 1$.

On a shelf, $4$ mathematics books and $3$ physics books are to be placed. If the books of the same subject must always be together, express the total number of possible arrangements using the product of several factorials and calculate its final value.

Solve the equation $\frac{(x+1)!}{(x-1)!} - 5x = 12$, finding the possible values for $x$ and discarding any that do not belong to the natural numbers.

Hint: Remember that $(x+1)! = (x+1) \cdot x \cdot (x-1)!$.

Simplify the algebraic expression $\frac{n!}{(n-2)!} + \frac{(n+1)!}{(n-1)!}$ as far as possible, expressing the final result as a polynomial in terms of $n$.

Six friends go to a restaurant for dinner and sit around a circular table. Given that at a circular table relative positions are what matter (circular permutation), express using factorials the number of different ways in which they can sit, and calculate the result.

Solve the equation $\frac{(n+2)!}{n!} = 72$ to find the value of the natural number $n$.

Prove the identity $\frac{n!}{(n-k)!} = n \cdot \frac{(n-1)!}{(n-k)!}$, detailing the algebraic steps for every natural number $n \geq k \geq 1$.

We are looking for an expression with factorials of the form $\dfrac{n!}{(n-k)!}$ whose numerical result is exactly equal to $210$, knowing also that when expanding and cancelling it, only 3 consecutive decreasing integer factors are obtained.

1. a) From the prime factorisation of the number $210$, deduce with reasoning what those 3 consecutive factors are and determine the exact numerical values of the natural numbers $n$ and $k$.
2. b) Verify that your expression $\dfrac{n!}{(n-k)!}$ is indeed equivalent to $210$.
3. c) Create and invent the statement of an everyday life problem (for example, in a school, sports, or commercial context) whose mathematical resolution corresponds exactly to the expression you found, explaining what both the number $n$ and the number $k$ represent in your story.

A Correos delivery van in the historic centre of Seville must deliver packages to $n$ different shops, stopping exactly once at each of them before returning to the logistics centre.

1. a) If on the first route of the day it must visit 6 shops, express the number of possible itineraries using a factorial and calculate it.
2. b) For the second route, one more shop is added, so it must visit 7 shops. Using the property $(n+1)! = (n+1) \cdot n!$, explain how to calculate the new number of routes from the result of the previous section, without needing to multiply the 7 factors one by one.
3. c) The logistics centre server uses an elementary algorithm that checks each of the possible routes to find the fastest one, taking $0,002$ seconds to evaluate each individual route. If on a special sales day the van had to visit 14 shops, knowing that $14! \approx 8,72 \cdot 10^{10}$, calculate how many whole days the computer would take to evaluate all the routes and justify why this search method is not feasible in real-world logistics management.