Each set X, contains 5 elements and each set Y, contains 2 elemen… - Vidyadip

Question

Each set $X,$ contains $5$ elements and each set $Y,$ contains $2$ elements and $\bigcup^\limits{20}_{\text{r=1}}\text{X}_\text{r}=\text{S =}\bigcup\limits^\text{n}_\text{r=1}\text{Y}_\text{r}.$ If each element of $S$ belongs to exactly $10$ of the $\text{X}'^\text{s}_\text{r}$ and to exactly $4$ of $\text{Y}'^\text{s}_\text{r}$, then find the value of $n.$

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Answer

Since each $X_r $ has $5$ elements and each element of S belong to exactly $10$ of $\text{X}'^\text{s}_\text{r}.$
$\therefore\text{ S}=\bigcup\limits^{20}_\text{r=1}\text{X}_\text{r}\Rightarrow\frac{1}{10}\sum\limits^{20}_\text{r=1}\text{n(X}_\text{r})=\frac{1}{10}(5\times20)=10.....\text{(i)}$
Since each $Y_r $ has $2$ elements and each element of $S$ belong to exactly $4$ of $\text{Y}'^\text{s}_\text{r}.$
$\therefore\text{ S}=\bigcup\limits^\text{n}_\text{r=1}\text{Y}_\text{r}\Rightarrow\frac{1}{4}\sum\limits^\text{n}_\text{r=1}\text{n(Y}_\text{r})=\frac{1}{4}\text{(2n)}=\frac{\text{n}}{2}.....\text{(ii)}$
From (i) and (ii), we get
$10=\frac{\text{n}}{2}\Rightarrow\text{n}=20.$

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