To find the total number of numbers that are divisible by 2 or 3 between 100 and 200 (both inclusive), we can use the concept of the inclusion-exclusion principle.

First, let's find the number of numbers divisible by 2 between 100 and 200. The first number divisible by 2 in this range is 100, and the last number divisible by 2 is 200. We can find the count of numbers divisible by 2 by finding the number of odd numbers in this range. The number of odd numbers between 100 and 200 (both inclusive) can be calculated as: [ \frac{{\text{{last number}} - \text{{first number}}}}{2} + 1 = \frac{{200 - 100}}{2} + 1 = 51 ]

Next, let's find the number of numbers divisible by 3 between 100 and 200. The first number divisible by 3 in this range is 102, and the last number divisible by 3 is 198. We can find the count of numbers divisible by 3 by dividing the range by 3 and adding 1. The number of numbers divisible by 3 between 100 and 200 (both inclusive) can be calculated as: [ \frac{{\text{{last number}} - \text{{first number}}}}{3} + 1 = \frac{{198 - 102}}{3} + 1 = 33 ]

However, we need to consider that some numbers are counted twice since they are divisible by both 2 and 3. To find the count of these numbers, we can divide the range by the least common multiple of 2 and 3, which is 6, and add 1. The number of numbers divisible by both 2 and 3 between 100 and 200 (both inclusive) can be calculated as: [ \frac{{\text{{last number}} - \text{{first number}}}}{6} + 1 = \frac{{198 - 102}}{6} + 1 = 17 ]

Now, let's apply the inclusion-exclusion principle to find the total count of numbers divisible by 2 or 3. We can calculate this by adding the count of numbers divisible by 2 to the count of numbers divisible by 3 and subtracting the count of numbers divisible by both 2 and 3. Total count of numbers divisible by 2 or 3 between 100 and 200 (both inclusive) can be calculated as: [ 51 + 33 - 17 = 67 ]

Therefore, the correct answer is option B) 67.

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To find the algebraic sum of the rest of the cards after removing Jack, Queen, King, and Ace, we need to consider the numerical values assigned to each card.

In a standard deck of cards, Jack, Queen, and King are assigned the values of 11, 12, and 13 respectively. Ace is assigned the value of 1.

The remaining cards in the deck are the number cards from 2 to 10. These cards have numerical values equivalent to their face value.

So, the algebraic sum of the rest of the cards can be calculated as:

2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 54

Therefore, the correct answer is C) 216.

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To calculate the number of small cubes that remain unpainted, we need to determine the number of small cubes on the surface of the big cube.

The big cube has dimensions of 12 mm on each side. Since it is made up of smaller cubes with dimensions of 3 mm, we can divide each side of the big cube by the side length of the small cube to find the number of small cubes along each edge.

12 mm / 3 mm = 4 small cubes

Since there are 4 small cubes along each edge of the big cube, the total number of small cubes on the surface can be found by multiplying the number of small cubes on one face by 6 (since there are 6 faces on a cube).

Number of small cubes on one face = 4 small cubes * 4 small cubes = 16 small cubes

Total number of small cubes on the surface = 16 small cubes * 6 faces = 96 small cubes

Since the big cube is painted on all its sides, the number of small cubes that remain unpainted is the total number of small cubes minus the number of small cubes on the surface.

Number of unpainted small cubes = Total number of small cubes - Number of small cubes on the surface Number of unpainted small cubes = 64 small cubes - 96 small cubes Number of unpainted small cubes = -32 small cubes

Since we cannot have a negative number of small cubes, it seems that there is an error in the given information or answer options. Please double-check the question and answer options.

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