A box contains tickets numbered 1 to N. n tickets are drawn from the box with replacement. The probability that the largest number on the tickets is k, is

Options

A. `((k)/(N))^(n)`
B. `((k-1)/(N))^(n)`
C. 0
D. None of these

Answer

To solve the problem of finding the probability that the largest number on the tickets drawn is \( k \), we can follow these steps:

### Step 1: Understand the Total Outcomes
When drawing \( n \) tickets from a box containing tickets numbered from \( 1 \) to \( N \) with replacement, the total number of possible outcomes is given by:
\[ N^n \]
This is because for each of the \( n \) draws, there are \( N \) possible tickets to choose from.

**Hint:** Remember that with replacement means each ticket can be drawn multiple times.

### Step 2: Determine Favorable Outcomes for Largest Number \( k \)
To ensure that the largest number drawn is exactly \( k \), we need to consider two conditions:
1. At least one ticket must be \( k \).
2. All other tickets must be from the set \( \{1, 2, \ldots, k\} \).

The total number of ways to draw tickets such that all drawn tickets are from \( \{1, 2, \ldots, k\} \) is:
\[ k^n \]
However, this count includes cases where \( k \) is not the largest number. We need to exclude those cases.

### Step 3: Exclude Cases Where \( k \) is Not the Largest
If \( k \) is not the largest, then all drawn tickets must be from the set \( \{1, 2, \ldots, k-1\} \). The number of ways to draw tickets such that all drawn tickets are from \( \{1, 2, \ldots, k-1\} \) is:
\[ (k-1)^n \]

### Step 4: Calculate Favorable Outcomes
Thus, the number of favorable outcomes where the largest number is exactly \( k \) can be calculated by:
\[ \text{Favorable outcomes} = k^n - (k-1)^n \]

### Step 5: Calculate the Probability
Now, the probability that the largest number on the tickets drawn is \( k \) is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{largest number is } k) = \frac{k^n - (k-1)^n}{N^n} \]

### Final Answer
Thus, the probability that the largest number on the tickets drawn is \( k \) is:
\[ \boxed{\frac{k^n - (k-1)^n}{N^n}} \]