A salesperson needs to visit 15 different offices during the week. In how many ways could she visit 3 different offices each day from Monday to Friday?
Well, if the salesperson needs to visit 3 different offices each day from Monday to Friday, we can use a combinatorics approach to count the possibilities.
On Monday, she can choose 3 offices out of the 15. That's 15 choose 3, or C(15, 3).
On Tuesday, she can choose another 3 offices out of the remaining 12, since she has already visited 3 on Monday. That's 12 choose 3, or C(12, 3).
She can continue this pattern for Wednesday, Thursday, and Friday, choosing 3 offices each day out of the remaining available options.
To calculate the total number of possibilities, we need to multiply all the choices together:
C(15, 3) * C(12, 3) * C(9, 3) * C(6, 3) * C(3, 3)
But since you asked for humor, let me put it in a clown way: Imagine the salesperson is juggling 15 different balls, each representing an office. Every day, she throws 3 balls in the air and catches them to visit those offices. On Monday, she throws the balls, and out of the 15, catches 3. On Tuesday, she throws the remaining 12 balls, and catches 3. She repeats this juggling act for Wednesday, Thursday, and Friday, with fewer balls each time. Now, if you count all the different ways she can catch those balls, that's the answer you're looking for! Tada!
To calculate the number of ways the salesperson can visit 3 different offices each day from Monday to Friday, we need to use the combination formula.
The formula to calculate the combination is:
C(n, r) = n! / (r! * (n-r)!)
Where:
- n is the total number of offices to visit (15 in this case).
- r is the number of offices the salesperson visits each day (3 in this case).
Now, let's calculate it step by step:
1. Calculate the total number of ways to choose 3 offices from 15.
- C(15, 3) = 15! / (3! * (15-3)!)
- C(15, 3) = 15! / (3! * 12!)
2. Simplify the equation.
- C(15, 3) = (15 * 14 * 13 * 12!) / (3! * 12!)
- C(15, 3) = (15 * 14 * 13) / (3!)
3. Calculate the factorial of 3.
- 3! = 3 * 2 * 1
- 3! = 6
4. Substitute the values back into the equation.
- C(15, 3) = (15 * 14 * 13) / 6
- C(15, 3) = 455
Therefore, there are 455 different ways the salesperson can visit 3 different offices each day from Monday to Friday.
To determine the number of ways the salesperson can visit 3 different offices each day from Monday to Friday, we need to use the concept of permutations.
First, we need to choose 3 different offices out of a total of 15 offices. This can be done using the combination formula:
C(n, r) = n! / ((n-r)! * r!)
Where n is the total number of offices and r is the number of offices we want to choose.
In this case, n = 15 and r = 3. Plugging these values into the formula, we get:
C(15, 3) = 15! / ((15 - 3)! * 3!)
= 15! / (12! * 3!)
Now, we need to consider the number of ways the selected 3 offices can be assigned to the 5 days of the week (Monday to Friday). This can be calculated using permutations.
P(n, r) = n! / (n - r)!
Where n is the total number of days (5) and r is the number of days we want to assign the offices (3). Plugging in the values:
P(5, 3) = 5! / (5 - 3)!
= 5! / 2!
Now, to find the total number of ways the salesperson can visit 3 different offices each day from Monday to Friday, we multiply the combination and permutation values:
C(15, 3) * P(5, 3)
= (15! / (12! * 3!)) * (5! / 2!)
Calculating this expression will give us the final answer.
Assuming the order within the day does not matter.
Monday: 15C3 ways
Tuesday: 12C3 ways
Wednesday: 9C3 ways
Thursday: 6C3 ways
Friday: 3C3 ways
Total= product of the above numbers,
=455*220*84*20*1
=168168000