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The problem involves a Poisson distribution with a mean of 20, where \( X \) is the number of pens sold by a retailer in a week, and the retailer earns a profit of Rs. 1 for each pen sold. We are asked to find the probability \( P(Y \geq 60) \), where \( Y \) represents the net profit, meaning that \( Y = X \) (since each pen sold gives a Rs. 1 profit).

### Step-by-step breakdown:

1. **Poisson Distribution:**
   - \( X \sim \text{Poisson}(\lambda = 20) \), where the mean number of pens sold is 20.
   - The Poisson probability mass function is:
   \[
   P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad \text{where} \quad \lambda = 20
   \]

2. **Objective:**
   - We are tasked to compute \( P(Y \geq 60) \), which is equivalent to \( P(X \geq 60) \), because \( Y = X \).

3. **Cumulative Probability:**
   - The cumulative probability \( P(X < 60) \) can be computed as:
   \[
   P(X < 60) = \sum_{y=0}^{59} \frac{e^{-20} 20^y}{y!}
   \]
   - Therefore, the probability we are looking for is:
   \[
   P(X \geq 60) = 1 - P(X < 60) = 1 - \sum_{y=0}^{59} \frac{e^{-20} 20^y}{y!}
   \]

### Answer:
Thus, the correct option is:

\[
\boxed{1 - \sum_{y=0}^{59} \frac{e^{-20} 20^y}{y!}}
\]
This matches option (c).

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Would you like a more detailed breakdown of the steps involved or further clarifications?

Here are 5 related questions you can explore:
1. What is the probability that exactly 20 pens are sold in a week?
2. How do you calculate the cumulative probability for a Poisson distribution?
3. What is the relationship between the mean and variance in a Poisson distribution?
4. How does the Poisson distribution apply to real-world scenarios?
5. Can you derive the probability mass function of a Poisson distribution?

**Tip:** When using the Poisson distribution, remember that the mean and variance are equal, which simplifies analysis in many problems.

Suppose that a retailer has 50 pens in stock and let X denote the number of pens that the retailer sells in a week. Assume that X follows the Poisson distribution with mean 20 and suppose that the retailer makes a profit of Rs. 1/- on each pen sold. Let Y denote the net profit made by the retailer during a week. Then, the value of P(Y ≥ 60) equals?

Learn how to calculate the probability P(Y ≥ 60) for a retailer's weekly profit using the Poisson distribution with a mean of 20. Understand the Poisson formula and cumulative probability approach, perfect for advanced high school or college students.

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