Question

There are 'n' locks and 'n' matching keys. If all the locks and keys are to be perfectly matched then find the maximum number of trials:

Answer 1

Hint: Permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.
The formula for permutations is: $\mathrm{^nP_r}=\dfrac{\mathrm{n} !} {(\mathrm{n}-\mathrm{r}) !}$
The formula for combinations is: $\mathrm{^nC_r}=\dfrac{\mathrm{n!}} {[\mathrm{r} !(\mathrm{n}-\mathrm{r}) !]}$
A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, we can select the items in any order. Combinations can be confused with permutations. In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.

Complete step-by-step answer:
Permutation of n different objects (when repetition is not allowed) Repetition, where repetition is allowed. Permutation when the objects are not distinct (Permutation of multi sets). When they refer to permutations, statisticians use a specific terminology. They describe permutations as n distinct objects taken r at a time. Which means n refers to the number of objects from which the permutation is formed; and r refers to the number of objects used to form the permutation.
Fer the first key maximum no. of trials $=n$ For the second key. $=(n-1)$ .
Generally, for ${{r}^{th}}$ key the trails will be: $(n-r+1)$
Then the maximum no. of trials is:
$T=n+(n-1)+(n-2)+................+1$
Here n is the first term and 1 is the last term.
$T=\frac{n}{2}$ (which signifies the addition of the first and the last term)
$T=\frac{n(n+1)}{2}=n+1$
$T=\frac{n(n+1)}{2}{{=}^{n+1}}{{C}_{2}}$
Hence the maximum no. of trials needed is $^{n+1}{{C}_{2}}$.

Note: One could say that a permutation is an ordered combination. The number of permutations of $\mathrm{n}$ objects taken $\mathrm{r}$ at a time is determined by the following formula:
$\mathrm{P}(\mathrm{n}, \mathrm{r})=\dfrac {\mathrm{n} !}{(\mathrm{n}-\mathrm{r}) !* \mathrm{n} !}$ is read $\mathrm{n}$ factorial and means all numbers from 1 to $\mathrm{n}$ multiplied. Combinations are a way to calculate the total outcomes of an event where order of
the outcomes do not matter.
To calculate combinations, we will use the formula $\mathrm{^nC_r}=\dfrac{\mathrm{n!}} {\mathrm{r!}^{*}(\mathrm{n}-\mathrm{r}) !}$ where $\mathrm{n}$ represents the total number of items, and $\mathrm{r}$ represents the number of items being chosen at a time. Thus, $\operatorname{^nP_r}(\mathrm{n}, \mathrm{r})$ The number of possibilities for choosing an ordered set of $\mathrm{r}$ objects $(\mathrm{a}$ permutation) from a total of n objects. Definition: $\operatorname{^nP_r}(\mathrm{n}, \mathrm{r})=\dfrac{\mathrm{n} !} {(\mathrm{n}-\mathrm{r}) ! \mathrm{^nC_r}(\mathrm{n}, \mathrm{r})}$.