4200 people attended a concert. From the 4200 available seating positions, 5 places have been reserved for randomly selected attendees labelled 1 to 5 (2nd row from the stage) In how many different ways can the 5 seating position be allocated to the 4200 attendees

To determine the number of different ways to allocate 5 specific seating positions to 4200 attendees, we can use the concept of permutations since the order in which the seats are assigned matters.

We start with 4200 attendees, and we need to select 5 of them to assign to the 5 reserved seats. 

The number of ways to choose and arrange 5 attendees from 4200 is given by the permutation formula:

\[ P(n, k) = \frac{n!}{(n-k)!} \]

where \( n \) is the total number of items (attendees), and \( k \) is the number of items to choose (seats).

Here, \( n = 4200 \) and \( k = 5 \).

Thus, the number of different ways to allocate the 5 seating positions is:

\[ P(4200, 5) = \frac{4200!}{(4200-5)!} = \frac{4200!}{4195!} \]

This simplifies to:

\[ P(4200, 5) = 4200 \times 4199 \times 4198 \times 4197 \times 4196 \]

Now let's calculate this product:

\[ 4200 \times 4199 \times 4198 \times 4197 \times 4196 \]

Performing the multiplication:

\[ 4200 \times 4199 = 17635800 \]
\[ 17635800 \times 4198 = 74138710400 \]
\[ 74138710400 \times 4197 = 311058144288000 \]
\[ 311058144288000 \times 4196 = 1306055307657088000 \]

So, the number of different ways to allocate the 5 seating positions to the 4200 attendees is \( 1,306,055,307,657,088,000 \).

Would you like more details or have any questions? 

Here are 5 more questions you might find interesting:
1. How many ways can 5 different books be arranged on a shelf?
2. How many ways can a committee of 3 be selected from 10 people?
3. How many permutations are there of the word "PERMUTATION"?
4. How many different ways can 8 people be seated at a round table?
5. How many ways can you distribute 7 identical balls into 4 distinct boxes?

**Tip:** When dealing with large factorials or permutations, breaking down the calculation step-by-step can help simplify the process and reduce errors.

Learn how to calculate permutations for allocating 5 reserved seats out of 4200 attendees. This problem involves combinatorial mathematics and permutations, where the order of selection matters. Discover the step-by-step calculation and understand the permutation formula P(n, k) = n! / (n-k)!.

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