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**Hint:**Use the following formulae- ${}^n{P_r} = \dfrac{{n!}}{{n - r!}}$ and ${}^n{C_r} = \dfrac{{n!}}{{r!n - r!}}$ where n=total number of things

And r=number of things to be selected. Put the given values and solve for n.

**Complete step-by-step answer:**Given, ${}^n{P_r} = 840$- (i)

And ${}^n{C_r} = 35$- (ii)

We have to find the value of n.

We know that- ${}^n{P_r} = \dfrac{{n!}}{{n - r!}}$ where n=total number of things and r=number of things to be selected.

On putting the given value we get in the formula we get,

$ \Rightarrow \dfrac{{n!}}{{n - r!}} = 840$ - (iii)

Also ${}^n{C_r} = \dfrac{{n!}}{{r!n - r!}}$ where n=total number of things and r=number of things to be selected.

On putting the value in this formula we get,

$ \Rightarrow \dfrac{{n!}}{{r!n - r!}} = 35$ - (iv)

On substituting the value from eq. (iii) to eq. (iv), we get,

$ \Rightarrow \dfrac{{840}}{{r!}} = 35$

On cross multiplication we get,

$ \Rightarrow $ $r! = \dfrac{{840}}{{35}}$

On dividing the numerator by denominator, we get

$ \Rightarrow r! = 24$

We can break $24$ into its factors then,

$ \Rightarrow r! = 4 \times 3 \times 2 \times 1 = 4!$

This means that r=$4$

On substituting the value of r in eq. (i)

$ \Rightarrow \dfrac{{n!}}{{n - 4!}} = 840$

On opening factorial of numerator we get,

$ \Rightarrow \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)\left( {n - 4} \right)!}}{{n - 4!}} = 840$

On solving we get,

$ \Rightarrow n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right) = 840$

On breaking $840$ into factors, we get

$ \Rightarrow n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right) = 42 \times 20$

On further breaking the factors we get,

$ \Rightarrow n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right) = 7 \times 6 \times 5 \times 4$

By observing the above equation we can see that the left hand side become equal to right hand side only when-

$ \Rightarrow n = 7$

**Hence, the correct option is ‘B’.**

**Note:**We can also find the value of r in above question using the formula-

$ \Rightarrow {}^n{P_r} = {}^n{C_r} \times r!$

We can directly obtain value of r by putting the given values-

$ \Rightarrow r! = \dfrac{{840}}{{35}} = 24$

We can then solve the question in the same manner as we solved in the above solution.

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