Let $A$ be a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen, and the set $A$ is reconstructed by replacing the elements of $P$. A subset $Q$ of $A$ is chosen again. The number of ways of choosing $P$ and $Q$ such the $Q$ contains just one element more than $P$ is

Question

Let $A$ be a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen, and the set $A$ is reconstructed by replacing the elements of $P$. A subset $Q$ of $A$ is chosen again. The number of ways of choosing $P$ and $Q$ such the $Q$ contains just one element more than $P$ is, ${ }^{2 n} C_{n-1}$, ${ }^{2 n} C_n$, ${ }^{2 n} C_{n+2}$, $2^{2 n+1}$

MARKS App · Accepted Answer

Hint : Required number of ways $\begin{aligned} & ={ }^{\mathrm{n}} \mathrm{C}_0 \cdot{ }^{\mathrm{n}} \mathrm{C}_1+{ }^{\mathrm{n}} \mathrm{C}_1 \cdot{ }^{\mathrm{n}} \mathrm{C}_2+\ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}-1} \cdot{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \ & ={ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}-1} \end{aligned}$

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