GATE CS 2019 Q.15 (CS Q.5): Combinatorics, Counting, Recurrence Relations and Generating Functions

Let $U = \{1, 2, \dots, n\}$ . Let $A = \{(x, X) \mid x \in X, X \subseteq U\}$ . Consider the following two statements on $|A|$ .

I. $|A| = n 2^{n-1}$
II. $|A| = \sum_{k=1}^n k \binom{n}{k}$

Which of the above statements is/are TRUE?

Only I

Only II

Both I and II

Neither I nor II

Official answer and solution

Official answer: C
From: Official GATE 2019 answer key

Solution

Count the elements of set A using double counting: first element-wise, then subset-size-wise.

Set $A$ consists of all ordered pairs $(x, X)$ such that $X \subseteq U$ and $x \in X$ , where $U = \{1, 2, \dots, n\}$ .

Evaluating Statement I (counting element by element):
For a fixed element $x \in U$ , how many subsets $X \subseteq U$ contain $x$ ? Since the remaining $n-1$ elements of $U$ can independently be either included or excluded, there are $2^{n-1}$ such subsets. Since there are $n$ distinct choices for $x \in U$ :

|A| = \sum_{x \in U} 2^{n-1} = n 2^{n-1}

Thus, Statement I is TRUE.

Evaluating Statement II (counting by subset size):
Let $k = |X|$ be the size of the subset $X$ , where $1 \le k \le n$ . There are $\binom{n}{k}$ subsets of size $k$ . For each such subset $X$ , there are $k$ choices of element $x \in X$ . Therefore, the total number of pairs is:

|A| = \sum_{k=1}^n k \binom{n}{k}

By standard binomial identities, $\sum_{k=1}^n k \binom{n}{k} = n 2^{n-1}$ . Thus, Statement II is also TRUE.

Both statements I and II are TRUE.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2019 Computer Science and Information Technology (CS)
Question: 15 of 65, printed as CS Q.5
Section: Computer Science and Information Technology

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