A train is going from cambridge to london stops at nine intermediate stations. Six persons enter the train during the journey with six different tickets. How many different sets of tickets they have had?

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Determine the Total Number of Tickets
The train stops at 9 intermediate stations. Each ticket corresponds to a journey between two stations. The number of different tickets can be calculated as follows:

- From Cambridge to the first station, there are 9 possible tickets (to any of the 9 stations).
- From the first station to the second station, there are 8 remaining tickets.
- Continuing this way, the number of tickets decreases by one for each subsequent station.

Thus, the total number of different tickets can be calculated as:
\[ \text{Total tickets} = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 \]

### Step 2: Calculate the Sum of Tickets
The sum of the first n natural numbers can be calculated using the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
In our case, \( n = 9 \):
\[ \text{Total tickets} = \frac{9 \times (9 + 1)}{2} = \frac{9 \times 10}{2} = 45 \]

### Step 3: Choose 6 Different Tickets
Now that we have determined that there are 45 different tickets, we need to find out how many different sets of 6 tickets can be chosen from these 45 tickets. This can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
where \( n \) is the total number of tickets (45) and \( r \) is the number of tickets to choose (6).

Thus, we need to calculate:
\[ \binom{45}{6} = \frac{45!}{6!(45 - 6)!} = \frac{45!}{6! \cdot 39!} \]

### Step 4: Simplify the Combination
To simplify \( \binom{45}{6} \):
\[ \binom{45}{6} = \frac{45 \times 44 \times 43 \times 42 \times 41 \times 40}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \]

### Step 5: Calculate the Value
Now, calculate the numerator and denominator:
- Numerator: \( 45 \times 44 \times 43 \times 42 \times 41 \times 40 \)
- Denominator: \( 720 \) (which is \( 6! \))

Calculating the numerator:
- \( 45 \times 44 = 1980 \)
- \( 1980 \times 43 = 85140 \)
- \( 85140 \times 42 = 3575880 \)
- \( 3575880 \times 41 = 146601960 \)
- \( 146601960 \times 40 = 5864078400 \)

Now divide by the denominator:
\[ \frac{5864078400}{720} = 8145060 \]

### Final Answer
Thus, the total number of different sets of tickets that the 6 persons can have is:
\[ \boxed{8145060} \]

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