+302 Lamant has a box containing only red marbles, blue marbles and green marbles. He needs to select at least 24 marbles without replacement to be sure at least one of them is green. He needs to select at least 20 marbles without replacement to be sure at least 1 of them is red. He needs to select at least 27 marbles without replacement to be sure all three colors appear among the marbles selected. How many marbles are in the box?
+743 Let's analyze each condition to see what it tells us about the minimum number of marbles in the box.
Red Marbles: Lamant needs at least 20 marbles to guarantee at least one red marble. This means there could be a box with only 19 non-red marbles (blue and green combined).
Green Marbles: Similarly, at least 24 marbles are needed to guarantee at least one green marble. This implies there could be a box with only 23 non-green marbles (red and blue combined).
All Three Colors: Finally, 27 marbles are needed to ensure all three colors are present. This means there could be a box with only 26 marbles that don't have all three colors (a combination of red and blue only, or blue and green only, or red and green only).
Combining the Information:
Looking at these conditions, we see that the number of non-red marbles (blue and green combined) must be greater than 19 (from condition 1) but less than or equal to 23 (from condition 2).
Similarly, the number of non-green marbles (red and blue combined) must be greater than 23 (from condition 2) but less than or equal to 26 (from condition 3).
The only way to satisfy both constraints is if there are exactly 23 non-red marbles and exactly 23 non-green marbles. This means the box must have a total of:
Red marbles = Total - Non-red marbles = Total - 23
Green marbles = Total - Non-green marbles = Total - 23
Finding the Total:
Since the number of red and green marbles must be the same (both equal to Total - 23), the total number of marbles in the box (including red, blue, and green) must be a multiple of 2. The smallest multiple of 2 that satisfies both conditions is:
Total = Red marbles + Green marbles + Blue marbles
Total = (Total - 23) + (Total - 23) + Blue marbles
2 * Total = 46 + Blue marbles
Since the total needs to be a whole number, the number of blue marbles must be even (because multiplying by 2 gives an even number). The smallest even number that satisfies all the conditions is:
Blue marbles = 2
Therefore, the total number of marbles in the box is:
Total = (46 + Blue marbles) / 2
Total = (46 + 2) / 2
Total = 48 / 2
Lamant's box contains a total of 24 marbles.
