[Brilliant Blog] Combinations

Main post link -> http://blog.brilliant.org/2013/01/30/combinations/

Read the full article at the blog, feel free to ask questions, discuss the topic, or the test yourself questions below

Test yourself

How many ways can three different appetizers be chosen from a menu that has 10 choices?
In New York city, all the streets are arranged in a grid. A hospital is located 5 blocks east and 6 blocks north of an accident. How many ways are there to get there, if we only go 1 block north or 1 block east at each intersection?
An office with 30 people wants to have 4 teams of 5 players participate in a charity tournament. How many ways can the teams be made? Hint: Be very careful.
How many ordered integer solutions $(a, b, c, d)$ are there to $a + b + c + d = 20$ subject to $a \geq 1, b \geq 2, c \geq 3, d \geq 4$ ?
Winston must choose 4 courses for his final semester of school. He must take at least 1 science class and at least 1 arts class. If his school offers 4 science classes, 3 arts classes and 3 other classes, how many different choices for classes does he have?

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Problem 2 is 11C5

Zi Song Yeoh - 8 years, 4 months ago

The fourth answer should be 13C3 which is 13x12x11/6 which is 286.

Rohan Rao - 8 years, 4 months ago

sorry i made my calculation of first wrong the correct answer for 1 is 120 as 720/6

nasir afroze - 8 years, 4 months ago

For fifth what is the correct answer? i got 672 as my answer, $4*3*8*7$

Himanshu Srivastava - 7 years, 11 months ago

the answer of 1 is 720

The third should be 30C5 x 25C5 x 20C5 x 15C5. Four teams will be formed, with 5 players each. 10 people will be left over.

This is actually a very common mistake made (also in this week's problem set). Remember that the order of the teams doesn't matter.

According to your method, how many ways are there for an office of 3 people to pick 3 teams of 1 person each? There should clearly only be 1 possible way.

Calvin Lin Staff - 8 years, 4 months ago

I understand. What is the correct way to do this question?

@Rohan Rao – Think about how to solve it. You should have been exposed to enough ideas in the blog that you should know how to adapt your initial answer.

Is the answer $\binom{30}{20} \times \frac{(20!)}{(5!)^{4} \times (4!)}$

Answer for Question 1:720 ways because after you pick the first one,there are nine choices left.After you pick the second one there are 8 choices left so $10 \times 9 \times 8$ =720

Tan Li Xuan - 8 years, 4 months ago

The answer is 120 because those ways of choosing the three appetizers are counted many times (ABC, ACB, BAC, BCA, CAB, CBA are the same).

The answer is ${10 \choose 3}$ = $\frac {10*9*8}{3*2*1}$ = $120$

Diego Stucchi - 8 years, 4 months ago

Thanks for pointing that out.It should be $10 \times 9 \times 8$ = 720 / 6 = 120

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$