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**Hint:**Mole concept gives the relationship between the number of moles, weight and molar mass of the compound.

-It is the most convenient method to express the amount of a substance

-Molar mass is calculated by adding up the atomic masses of the element combined to form a molecule.

**Formula used:**

$n=\dfrac{W}{M}$

where, $n$ is the number of moles, $W$ is the weight of the compound and $M$ is the molar mass of the compound.

Number of molecules $=6.023\times {{10}^{23}}\times $ moles

**Complete step by step solution:**

The number of moles is defined as the weight per molar mass. It is a unit of measurement which is used to express the amount of reactants and products in a chemical reaction.

In this question, a gaseous mixture of oxygen and nitrogen is in the ratio of $1:4$ by weight.

Molar mass of oxygen $({{O}_{2}})$ is $32$ and that of nitrogen $({{N}_{2}})$ is $28$ .

The molar ratio of oxygen $({{O}_{2}})$ : nitrogen $({{N}_{2}})$ will be:

$\dfrac{{{W}_{{{O}_{2}}}}}{{{M}_{{{O}_{2}}}}}:\dfrac{{{W}_{{{N}_{2}}}}}{{{M}_{{{N}_{2}}}}}$

Now, substituting the values in the above formula, we get,

The molar ratio of oxygen $({{O}_{2}})$ : nitrogen $({{N}_{2}})$ will be:

$\dfrac{1}{32}:\dfrac{4}{28}$

On further solving, we get,

$\Rightarrow \dfrac{1}{32}:\dfrac{1}{7}$

This is equation $(1)$

To calculate the number of moles, formula used is:

$n=\dfrac{W}{M}$

where, $n$ is the number of moles, $W$ is the weight of the compound and $M$ is the molar mass of the compound.

Now, applying this formula in equation $(1)$ , we get,

Ratio of their molecules:

$\Rightarrow \dfrac{6.023\times {{10}^{23}}}{32}:\dfrac{6.023\times {{10}^{23}}}{7}$

On further solving, the final ratio would be,

$\Rightarrow 7:32$

**Therefore, the correct option is (C) $7:32$ .**

**Note:**

-Avogadro’s number is defined as the proportionality factor that tells the relationship between the number of constituent particles with the amount of substance in a sample. Its SI unit is $mo{{l}^{-1}}$ . It is denoted with a symbol, ${{N}_{A}}$ .

${{N}_{A}}=6.023\times {{10}^{23}}mo{{l}^{-1}}$

-It is the number of units in a mole of any substance.

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