Permutations And Combinations

To solve the problem, we need to determine the number of seating arrangements when 20 guests and 1 host are seated around a circular table, with the condition that two particular persons must sit on either side of the host.

1. Understanding the Problem:
- Total persons = 20 guests + 1 host = 21 persons
- Arrangement is circular with 2 specific guests (let's call them G₁ and G₂) required to sit adjacent to the host (H)

2. Fixing the Host's Position (Circular Permutation):
For circular arrangements, we fix one position to eliminate equivalent rotations:
- Fix the host in a specific seat: H = fixed position
- This converts the problem to linear arrangement relative to H

3. Arranging the Two Special Guests:
The condition requires:
- G₁ must sit immediately to H's left or right
- G₂ must sit on the opposite side
Possible arrangements:
1. (G₁-H-G₂)
2. (G₂-H-G₁)
Number of arrangements for these two guests:

4. Arranging Remaining Guests:
- After placing H, G₁, and G₂, we have 18 seats left
- 18 guests can be arranged in these seats in: ways

5. Calculating Total Arrangements:
Total valid arrangements = (Arrangements of G₁/G₂) × (Arrangements of others)

Verification:
- Host's position is fixed (1 way)
- 2 choices for adjacent guests' positions
- 18! arrangements for others
- Total matches our calculation

Final Answer:
The number of valid seating arrangements is .

To solve the problem, we need to find the number of 5-letter words that can be formed from the word "PULSE" without repeating any letters.

1. Understanding the Problem:
We are given the word "PULSE" which contains 5 distinct letters: P, U, L, S, E. We need to determine how many different 5-letter arrangements (words) can be made using each letter exactly once.

2. Recognizing the Mathematical Concept:
Since the order of letters matters and we cannot repeat any letters, this is a permutation problem. The number of possible arrangements is given by the factorial of the number of distinct items.

3. Applying the Permutation Formula:
For n distinct items, the number of possible permutations is n! (n factorial). Here, n = 5 (letters in "PULSE"), so we calculate 5!:

4. Verifying the Calculation:
We can also think step-by-step:
- 1st letter: 5 possible choices
- 2nd letter: 4 remaining choices
- 3rd letter: 3 remaining choices
- 4th letter: 2 remaining choices
- 5th letter: 1 remaining choice
Total arrangements = 5 × 4 × 3 × 2 × 1 = 120

Final Answer:
The number of 5-letter words that can be formed from "PULSE" without repetition is .