A bag contains 2 white marbles, 3 black marbles and 4 red marbles. Find in how…
2025
A bag contains 2 white marbles, 3 black marbles and 4 red marbles. Find in how many ways, 3 marbles can be drawn, so that at least one black marble is included in each draw?
Answer: A. 64 — To find the number of ways to draw 3 marbles such that at least one is black, it is easiest to subtract the cases where no black marbles are drawn from the…
- A.
64
- B.
67
- C.
89
- D.
98
Attempted by 2 students.
Show answer & explanation
Correct answer: A
To find the number of ways to draw 3 marbles such that at least one is black, it is easiest to subtract the cases where no black marbles are drawn from the total possible ways to draw 3 marbles.
Step-by-Step Solution
Calculate the Total Number of Marbles: Total marbles = 2 (white) + 3 (black) + 4 (red) = 9 marbles.
Total Ways to Draw 3 Marbles (Unrestricted): This is "9 choose 3" (calculated as (9 * 8 * 7) / (3 * 2 * 1)): 9 * 8 * 7 / 6 = 84 total combinations.
Calculate Ways to Draw 3 Marbles with NO Black Marbles: If we draw no black marbles, we must draw all 3 from the non-black pool (2 white + 4 red = 6 non-black marbles). This is "6 choose 3" (calculated as (6 * 5 * 4) / (3 * 2 * 1)): 6 * 5 * 4 / 6 = 20 combinations.
Calculate Ways with At Least One Black Marble: Total ways - Ways with no black marbles = 84 - 20 = 64.