**number of possible ways**for which a $5$-

**card poker hand**of a total of $52 $

**cards**should not have

**the queen of hearts**.

**In order to solve this problem the right way, we need to understand the concept of**

**Permutation**and

**Combination**. In mathematics, the definition of

**combination**is the

**arrangement**of its given members irrespective of their order. A

**permutation**in mathematics is the arrangement of its members in a

**definite order.**The formula of

**Combination**and

**Permutation**is as follows: \[C\left(n,r\right)=\frac{n!}{r! \left(n-r\right)!}\]

**$C\left(n,r\right)$ = number of combinations**

**$n$ = total number of objects**For this, we will use the formula of

**combination**and we have to find the

**total number of possible outcomes.**Then by fixing the one card from the five cards as the

**queen of hearts,**we have to again find the

**probability**with $n= 51$ and $r=4 $.

## Expert Answer

According to the**rule of complement,**we have: \[ P\ (\vec{E})\ =\ 1\ -\ P(E)\] For this we will use the formula of Combination as

**the order of the members**does not matter in the

**set of cards**containing $52 $

**cards**and $5$

**cards**were selected from the total of $52$

**cards**: \[C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}\] Here $n= 52 $ and $r = 5$ so the

**formula for combination**will be: \[ C\left(52, 5\right)=\frac{ 52!}{ 5! \left( 52 – 5 \right)!} \] Solving the above equation: \[ C\left(52, 5\right)=\frac{ 52!}{ 5! \times 47! } \] \[ C\left(52, 5\right) = 2,598,960 \] Now to find the

**number of hands**with the

**queen of hearts**in it. Let us suppose we have selected

**queen of hearts**as

**one card,**then the remaining will be $4$

**cards**left out of the $5 $

**cards**to be selected from the

**deck of $ 52$ cards**. We have $n= 51 $ and $r = 4$ so the

**formula for combination**will be: \[ C\left(51, 4\right)=\frac{ 51!}{ 4! \left( 51 – 4 \right)! } \] Solving the above equation: \[ C\left(51, 4\right)=\frac{ 51!}{ 4! \times 47! } \] \[ C\left(51, 4\right) = 249,900 \] Thus, the

**probability**of not finding the

**queen of hearts**in a

**five-card poker**from a total of $52$

**cards**will be the

**possibility of the card divided by the total number of outcomes**: \[ P(E) = \dfrac {249,900 }{ 2,598,960 } \] \[ P(E) = \dfrac {5 }{52 } \] Now

**putting the value**of $P$ in the equation: \[ P\ (\vec{E})\ =\ 1\ -\ P(E)\] \[ P\ (\vec{E})\ =\ 1\ -\ \dfrac {5}{52}\] \[ P\ (\vec{E})\ =\ \dfrac {47}{52}\] \[ P\ (\vec{E})=0.9038 \]

## Numerical Results

Thus, the**probability**of not finding the

**queen of hearts**in a

**five-card poker**from a total of $52$

**cards**will be: \[ P\ (\vec{E})\ =\ \dfrac {47}{ 52}\]

## Example

A group of $4$**players**is $P$, $Q$, $R$, $S$. In how many ways can a team of $2$

**members**be formed? Using

**the Combination formula:**$n=4$ $r=2$ \[C\left(4,2 \right)=\frac{4!}{2!\left(4-2\right)!}\] \[C\left(4,2 \right)=6\]

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