Each of the following question is followed by two statements I , and II. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question.
In how many ways can 4 boys and 5 girls be selected?
I. there are 20 persons (boys + girls) in the group out of which 12 are boys.
II. the ratio ofboys to girls in the group is 3:2.

Options

A. Only Statement I alone.
B. Only Statement II alone.
C. Both Statements I and II together.
D. Neither Statement I nor II is sufficient.

Answer

To determine how many ways we can select 4 boys and 5 girls from a group, we need to analyze the two statements provided.

**Step 1: Understand the question.**
We need to find out how many ways we can select 4 boys and 5 girls from a group.

**Hint:** Identify the total number of boys and girls in the group.

**Step 2: Analyze Statement I.**
Statement I states that there are 20 persons in total (boys + girls) in the group, out of which 12 are boys.
- From this, we can deduce that if there are 12 boys, then the number of girls is 20 - 12 = 8 girls.

**Hint:** Calculate the number of girls based on the information given in Statement I.

**Step 3: Calculate the combinations using Statement I.**
Now that we know there are 12 boys and 8 girls, we can calculate the number of ways to select 4 boys from 12 and 5 girls from 8.
- The number of ways to choose 4 boys from 12 is given by the combination formula: \( \binom{12}{4} \).
- The number of ways to choose 5 girls from 8 is given by: \( \binom{8}{5} \).

**Hint:** Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) to find the values.

**Step 4: Calculate the total combinations.**
The total ways to select 4 boys and 5 girls can be calculated as:
\[ \text{Total Ways} = \binom{12}{4} \times \binom{8}{5} \]

**Hint:** Multiply the results of the combinations to get the total number of ways.

**Step 5: Analyze Statement II.**
Statement II states that the ratio of boys to girls in the group is 3:2.
- This ratio does not provide specific numbers of boys and girls, so we cannot determine the exact counts needed to answer the question.

**Hint:** Consider whether a ratio alone can provide enough information to solve the problem.

**Step 6: Conclusion.**
From the analysis:
- Statement I is sufficient to answer the question as it provides the exact number of boys and girls.
- Statement II is not sufficient as it only provides a ratio without specific counts.

Thus, the answer is that only Statement I is necessary to answer the question.

**Final Answer:** Only Statement I is sufficient to answer the question.