Daily Practice

Solution

$6$ are blue and green - $b+g=6$ $8$ are red and blue - $r+b=8$ $4$ are red and green - $r+g=4$ We can do trial and error. Let's make blue $5$. That makes green $1$ and red $3$ because $6-5=1$ and $8-5=3$. To check this, let's plug $1$ and $3$ into $r+g=4$, which works. Now count the number of marbles - $5+3+1=9$. So the answer is $boxed\{textbf\{(C)\}\; 9\}.$

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

Solution

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}}

Solution

The total length of all of the arcs is $100pi\; +80pi\; +60pi=240pi$. Since we want the path from the center, the actual distance will be subtracted by $2pi$ 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 $4pi$ Therefore, the answer is $240pi-2pi=boxed\{textbf\{(A)\}\; 238pi\}$. ~PowerQualimit Video Solution: https://www.youtube.com/watch?v=zZGuBFyiQrk by WhyMath

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