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**Hint:**The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The probability formula is used to compute the probability of an event to occur. To recall, the likelihood of an event happening is called probability.

- Probability Rule One (For any event $A, 0 \leq P(A) \leq 1)$

- Probability Rule Two (The sum of the probabilities of all possible outcomes is 1 )

- Probability Rule Three (The Complement Rule)

- Probabilities Involving Multiple Events.

- Probability Rule Four (Addition Rule for Disjoint Events)

- Finding $\mathrm{P}(\mathrm{A}$ and $\mathrm{B})$ using Logic.

**Complete step-by-step answer:**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: $\mathrm{nPr}=\mathrm{n} ! /(\mathrm{n}-\mathrm{r}) !$

The formula for combinations is: $\mathrm{nCr}=\mathrm{n!} /[\mathrm{r} !(\mathrm{n}-\mathrm{r}) !]$

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.

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. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.

$\mathrm{C}\left(\mathrm{n}_{1} \mathrm{r}\right)+2 \mathrm{C}\left(\mathrm{n}_{1} \mathrm{r}-1\right)+\mathrm{C}(\mathrm{n}, \mathrm{r}-2)$

$={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+2^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-2}$

$={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-2} \ldots . .(1)$

Now we know that

${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}} \ldots . .(2)$

$\therefore \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}}$

${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-2}={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}-1}$

$\therefore(1)$ becomes

${ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}-1}={ }^{\mathrm{n}+2} \mathrm{C}_{\mathrm{r}}[$ By Property Used in (2)]

$\Rightarrow \mathrm{C}(\mathrm{n}+2, \mathrm{r})$

**Hence the correct option is D.**

**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:

$\mathrm{P}(\mathrm{n}, \mathrm{r})=\mathrm{n} !(\mathrm{n}-\mathrm{r}) ! \mathrm{n} !$ 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 $\mathrm{nCr}=\mathrm{n!} / \mathrm{r!}^{*}(\mathrm{n}-\mathrm{r}) !,$ where $\mathrm{n}$ represents the total number of items, and $\mathrm{r}$ represents the number of items being chosen at a time. Thus, $\operatorname{nPr}(\mathrm{n}, \mathrm{r})$ The number of possibilities for choosing an ordered set of $\mathrm{r}$ objects $(\mathrm{a}$ permutation) from a total of n objects. Definition: $\operatorname{nPr}(\mathrm{n}, \mathrm{r})=\mathrm{n} ! /(\mathrm{n}-\mathrm{r}) ! \mathrm{nCr}(\mathrm{n}, \mathrm{r})$.

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