In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?
Solution
are blue and green -
are red and blue -
are red and green -
We can do trial and error. Let's make blue . That makes green and red because and . To check this, let's plug and into , which works. Now count the number of marbles - . So the answer is
Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
Solution
Note: This solution uses the non-negative version for stars and bars. A solution using the positive version of stars is similar (first removing an apple from each person instead of 2).
This method uses the counting method of stars and bars (non-negative version). Since each person must have at least 2 apples, we can remove 2*3 apples from the total that need to be sorted. With the remaining 18 apples, we can use stars and bars to determine the number of possibilities. Assume there are 18 stars in a row, and 2 bars, which will be placed to separate the stars into groups of 3. In total, there are 18 spaces for stars + 2 spaces for bars, for a total of 20 spaces. We can now do 20 choose 2. This is because if we choose distinct 2 spots for the bars to be placed, each combo of 3 groups will be different, and all apples will add up to 18. We can also do this because the apples are indistinguishable. 20 choose 2 is 190, therefore the answer is boxed{textbf{(C) }190}.
~goofytaipan91
We rewrite: frac{1}{2}cdotleft(frac{3cdot2}{2cdot3}right)left(frac{4cdot3}{3cdot4}right)cdotsleft(frac{99cdot98}{98cdot99}right)cdotfrac{100}{99}
The middle terms cancel, leaving us with
left(frac{1cdot100}{2cdot99}right)= boxed{textbf{(B)}frac{50}{99}}
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are inches, inches, and inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
[figure]
Solution
The total length of all of the arcs is . Since we want the path from the center, the actual distance will be subtracted by because it's already half the circumference through semicircle A, which needs to go half the circumference extra through semicircle B, and it's already half the circumference through semicircle C, and the circumference is Therefore, the answer is .
~PowerQualimit
Video Solution:
https://www.youtube.com/watch?v=zZGuBFyiQrk by WhyMath
An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it
is an odd integer whose digits are all distinct?
Solution
There are 5 options for the last digit (1,3,5,7,9) as the integer must be odd. The first digit now has 8 options left (it can't be 0 or the same as the last digit). The second digit also has 8 options left (it can't be the same as the first or last digit). Finally, the third digit has 7 options (it can't be the same as the three digits that are already chosen).
Since there are 9,000 total integers, our answer is frac{8 cdot 8 cdot 7 cdot 5}{9000} = boxed{textbf{(B)} frac{56}{225}}.
==Video Solution (CREATIVE THINKING + ANALYSIS!!!)==
https://youtu.be/EI3SebxlOBs
~Education, the Study of Everything
==Video Solution==
https://youtu.be/4RsSWWXpGCo
https://youtu.be/tJm9KqYG4fU?t=3114
https://youtu.be/JmijOZfwM_A
~savannahsolver
https://www.youtube.com/watch?v=2G9jiu5y5PM ~David