NIMCET 2007 — Reasoning PYQ

Step 1: Breaking Down the Given Information

From the problem description, we can identify two main attributes for classification: Gender (Boys / Girls) and Hair Color (With Brown Hair / Without Brown Hair).

Let's look at the given values:

• Number of girls with brown hair = $7$

• Number of persons who do not have brown hair = $15$

The phrasing "$13$ boys and $7$ girls with brown hair" logically implies that there are $13$ boys in total in the group.

Step 2: Logical Set Overlapping for Minimization

To find the minimum total number of people in the group, we need to maximize the overlap between the existing categories.

The two independent parameters given that cannot overlap with each other are:

1. Girls with brown hair = $7$

2. Persons without brown hair = $15$

Since a person cannot simultaneously have brown hair and not have brown hair, these two groups are completely separate (disjoint sets):

$$\text{Base Group Size}=7+15=22\text{ persons}$$

Now, we have $13$ boys that must be accounted for in the group. To keep the total group size as small as possible, we should fit as many of these $13$ boys as possible into the existing categories.

• Can any of the $13$ boys be part of the "$7$ girls with brown hair"? No (boys cannot be girls).

• Can the $13$ boys be part of the "$15$ persons without brown hair"? Yes.

If we assume all $13$ boys do not have brown hair, they completely fit inside the set of $15$ non-brown-haired people:

$$\text{Boys without brown hair}=13$$

$$\text{Girls without brown hair}=15-13=2$$

Step 3: Calculating the Minimum Total Capacity

By overlapping the sets this way, no extra people need to be added to the group beyond our base disjoint sets.

$$\text{Minimum total persons}=(\text{Girls with brown hair})+(\text{Persons without brown hair})$$

$$\text{Minimum total persons}=7+15=22$$

Let's double-check the distribution to ensure all constraints are satisfied:

• Total Boys = $13$ (all have non-brown hair)

• Total Girls = $9$ ($7$ with brown hair + $2$ with non-brown hair)

• Total people with brown hair = $7$

• Total people without brown hair = $13\text{ boys}+2\text{ girls}=15$

Everything matches perfectly. The absolute minimum number of persons the group can have is $22$.

Step 4: Comparing with Options

Let's look at the options provided:

• (a) $13$

• (b) $26$

• (c) $25$

• (d) None

Since $22$ is not among choices (a), (b), or (c), it falls under option (d).

Final Answer

The correct option is (d) None.