Mathematics - Permutation & Combination Question with Solution | TestHub

MathematicsPermutation & CombinationnPr & nCrMedium2 minQB

MathematicsMediumsingle choice

Passage / Comprehension

A is a set containing ‘n’ elements. A subset P of A is chose at random. The set A is reconstructed by replacing the elements of ‘P’ A subset Q of ‘A’ is again chosen at random. Then the number of ways of selecting ‘P’ and Q such that

$P \cup Q = A$

Options:

Answer:

Solution:

Let 'P' be constructed by selecting 'r' elements from 'n' elements of set 'A'. This can be done in $^{n} c_{r}$ ways. In order to construct Q such that $P \cup Q = A$ We must include n-r elements of set A in Q and any number of elements from ' r ' elements selected for P This can be done in $1 \times (^{r} c_{0} +^{r} c_{1} +^{r} c_{2} + \dots . +^{r} c_{r}) = 2^{r}$ Number of ways of selecting P and Q such that $P ⋃ Q = A$ is $^{n} c_{r} (2^{r})$ but ' r ' varies from 0 to n $\Rightarrow reqd no.of ways = \sum_{r = 0}^{n}^{n} c_{r} 2^{r} = (1 + 2)^{n} = 3^{n}$

Stream:JEESubject:MathematicsTopic:Permutation & CombinationSubtopic:nPr & nCr

⏱ 2mℹ️ Source: QB

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