Key Concepts and Formulas

• Combinations: The number of ways to choose $r$ items from a set of $n$ items (where order doesn't matter) is given by the binomial coefficient: $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$
• Addition Principle: If there are $n$ mutually exclusive ways to do something, and $m$ mutually exclusive ways to do something else, then there are $n + m$ ways to do either.
• Multiplication Principle: If there are $n$ ways to do one thing and $m$ ways to do another, then there are $n \times m$ ways to do both.

Step-by-Step Solution

Step 1: Understand the problem and the constraint. We need to find the number of ways to draw 4 marbles from the urn such that at most 3 of them are red. This means we can have 0, 1, 2, or 3 red marbles. The urn contains 5 red, 4 black, and 3 white marbles, totaling 12 marbles.

Step 2: Calculate the number of ways to choose 4 marbles with 3 red marbles. We choose 3 red marbles from the 5 available and 1 marble from the remaining 7 non-red marbles (4 black + 3 white). The number of ways is: $\binom{5}{3} \times \binom{7}{1} = \frac{5!}{3!2!} \times \frac{7!}{1!6!} = \frac{5 \times 4}{2} \times 7 = 10 \times 7 = 70$

Step 3: Calculate the number of ways to choose 4 marbles with 2 red marbles. We choose 2 red marbles from the 5 available and 2 marbles from the remaining 7 non-red marbles. The number of ways is: $\binom{5}{2} \times \binom{7}{2} = \frac{5!}{2!3!} \times \frac{7!}{2!5!} = \frac{5 \times 4}{2} \times \frac{7 \times 6}{2} = 10 \times 21 = 210$

Step 4: Calculate the number of ways to choose 4 marbles with 1 red marble. We choose 1 red marble from the 5 available and 3 marbles from the remaining 7 non-red marbles. The number of ways is: $\binom{5}{1} \times \binom{7}{3} = \frac{5!}{1!4!} \times \frac{7!}{3!4!} = 5 \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 5 \times 35 = 175$

Step 5: Calculate the number of ways to choose 4 marbles with 0 red marbles. We choose 0 red marbles from the 5 available and 4 marbles from the remaining 7 non-red marbles. The number of ways is: $\binom{5}{0} \times \binom{7}{4} = \frac{5!}{0!5!} \times \frac{7!}{4!3!} = 1 \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 1 \times 35 = 35$

Step 6: Calculate the total number of ways. Since these cases are mutually exclusive, we add the number of ways for each case to get the total number of ways: $\text{Total} = 70 + 210 + 175 + 35 = 490$

Step 7: Consider the alternate approach - find total ways of choosing 4 marbles and subtract ways of choosing 4 red marbles. Total ways to choose 4 marbles from 12: $\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495$ Ways to choose 4 red marbles: $\binom{5}{4} = 5$ Ways to choose at most 3 red marbles: $495 - 5 = 490$

Step 8: Compare with the correct answer. The calculated answer 490 does not match the provided correct answer of 5. This indicates an error in the problem statement or the given answer. The problem asks for the number of ways to draw 4 marbles with at most 3 red marbles. Using the complementary approach, we calculate the total number of ways to choose 4 marbles from 12, then subtract the number of ways to draw 4 red marbles.

Total number of ways to choose 4 marbles from 12 is $\binom{12}{4} = \frac{12!}{4!8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495$. Number of ways to choose 4 red marbles is $\binom{5}{4} = \frac{5!}{4!1!} = 5$. Therefore, the number of ways to choose at most 3 red marbles is $495 - 5 = 490$. Since the provided "Correct Answer" is 5, and the calculated answer is 490, there must be an error in the question or provided answer. Let us assume the correct answer is 490.

Common Mistakes & Tips

• Remember to consider all possible cases when dealing with "at most" or "at least" type problems.
• Double-check your calculations, especially when dealing with factorials and combinations.
• Consider using complementary counting (total possibilities minus unwanted possibilities) when it simplifies the problem.

Summary

We calculated the number of ways to draw 4 marbles with at most 3 red marbles by considering the cases where we have 0, 1, 2, or 3 red marbles and summing the number of ways for each case. Alternatively, we can find the total number of ways to choose 4 marbles and subtract the number of ways to choose 4 red marbles. Both methods lead to the same result, 490. The given "Correct Answer" is incorrect. Assuming there is an error in the question or provided answer, the correct answer should be 490.

Final Answer

The final answer is \boxed{490}.