CBSE Class 11 Mathematics Practice Worksheet A on Fundamental Principle of Counting, including questions on selection, appointments, route counting, code words, and number formation.

Problem Analysis and Solutions

The worksheet provided is about the Fundamental Principle of Counting, which involves permutations, combinations, and basic counting principles. Let's solve each question step by step.

---

#### Q1:
Question: In a class there are twenty boys and 16 girls. The teacher wants to select either a boy or a girl to represent the class in a function. In how many ways can the teacher make his selection?

Solution:
- There are 20 boys, so the teacher has 20 choices if he selects a boy.
- There are 16 girls, so the teacher has 16 choices if he selects a girl.
- Since the teacher can choose either a boy or a girl, we use the addition principle:
\[
\text{Total ways} = \text{Number of boys} + \text{Number of girls} = 20 + 16 = 36
\]

Answer:
\[
\boxed{36}
\]

---

#### Q2:
Question: From among the 50 teachers in a school, one principal and two vice-principals are to be appointed. In how many ways can this be done?

Solution:
- First, select the principal. There are 50 choices for the principal.
- After selecting the principal, 49 teachers remain. Select 2 vice-principals from these 49 teachers. The number of ways to choose 2 vice-principals from 49 is given by the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):
\[
\binom{49}{2} = \frac{49 \times 48}{2 \times 1} = 1176
\]
- The total number of ways to appoint one principal and two vice-principals is the product of the number of ways to choose the principal and the number of ways to choose the vice-principals:
\[
\text{Total ways} = 50 \times 1176 = 58800
\]

Answer:
\[
\boxed{58800}
\]

---

#### Q3:
Question: The following figure represents the route chart connecting three stations \( P, Q, \) and \( R \). In how many ways can a person go from station \( P \) to station \( R \)?

Solution:
- The problem does not provide the figure, but typically such questions involve counting paths between nodes in a graph. Assuming a standard setup where there are multiple routes from \( P \) to \( R \) via \( Q \), we would count all possible paths.
- Without the figure, we cannot provide a specific numerical answer. However, the general approach would be to list all possible routes and count them.

Note: Provide the figure for an exact solution.

---

#### Q4:
Question: A code word is to consist of a single English alphabet followed by two distinct numbers between 1 to 9. For example, \( N34 \) is a code word. How many such code words are there? How many of them have an even integer?

Solution:
- Step 1: Total number of code words
- There are 26 choices for the English alphabet (A to Z).
- There are 9 choices for the first digit (1 to 9).
- There are 8 choices for the second digit (since it must be distinct from the first digit).
- By the multiplication principle:
\[
\text{Total code words} = 26 \times 9 \times 8 = 1872
\]

- Step 2: Code words with an even integer
- Even digits between 1 and 9 are: 2, 4, 6, 8 (4 choices).
- Case 1: The first digit is even.
- 4 choices for the first digit.
- 8 choices for the second digit (it can be any digit except the first).
- Total for this case: \( 26 \times 4 \times 8 = 832 \).
- Case 2: The second digit is even.
- 5 choices for the first digit (odd digits: 1, 3, 5, 7, 9).
- 4 choices for the second digit (even digits).
- Total for this case: \( 26 \times 5 \times 4 = 520 \).
- Total code words with at least one even digit:
\[
832 + 520 = 1352
\]

Answers:
- Total code words: \( \boxed{1872} \)
- Code words with an even integer: \( \boxed{1352} \)

---

#### Q5:
Question: A code word is to consist of a single English alphabet followed by two distinct numbers between 1 to 9. For example, \( N34 \) is a code word. How many such code words are there? How many of them have an even integer?

Solution:
- This question is identical to Q4. The solution remains the same.

Answers:
- Total code words: \( \boxed{1872} \)
- Code words with an even integer: \( \boxed{1352} \)

---

#### Q6:
Question: Find the number of 4-letter words, with or without meaning, which can be formed out of the letters of the word "ROSE," where the repetition of the letters is not allowed.

Solution:
- The word "ROSE" has 4 distinct letters: R, O, S, E.
- We need to form 4-letter words using these 4 letters without repetition.
- The number of ways to arrange 4 distinct letters is given by the permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n = 4 \) and \( r = 4 \):
\[
P(4, 4) = 4! = 4 \times 3 \times 2 \times 1 = 24
\]

Answer:
\[
\boxed{24}
\]

---

#### Q7:
Question: How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

Solution:
- A 3-digit even number must end in an even digit. The even digits available are: 2, 4, 6 (3 choices).
- For the hundreds place, any of the 6 digits can be used (1, 2, 3, 4, 5, 6).
- For the tens place, any of the 6 digits can be used.
- For the units place, only the 3 even digits can be used.
- By the multiplication principle:
\[
\text{Total 3-digit even numbers} = 6 \times 6 \times 3 = 108
\]

Answer:
\[
\boxed{108}
\]

---

Final Answers:

1. \( \boxed{36} \)
2. \( \boxed{58800} \)
3. Figure needed for exact solution.
4. Total code words: \( \boxed{1872} \); Code words with an even integer: \( \boxed{1352} \)
5. Total code words: \( \boxed{1872} \); Code words with an even integer: \( \boxed{1352} \)
6. \( \boxed{24} \)
7. \( \boxed{108} \)