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\(\begin{align*} \binom{7}{3}\binom{6}{2} + \binom{7}{4}\binom{6}{1} + \binom{7}{5} &= 35 \cdot 15 + 35 \cdot 6 + 21 \\ &= 7 \cdot (75+30+3) \\ &= 7 \cdot 108 \\ &= \boxed{756} \end{align*}\)

Guess and check.

If every batch is fudge: 9x3=27

If every batch is peanut butter: 9x10=90

Each time we replace a peanut butter batch with a fudge batch, 7 brownies are deducted from the total. 

90-62=28 difference between all PB and actual amount

28/7=4 batches of fudge

9-4=5 batches of PB

There are 4 batches of fudge and 5 batches of PB.

(doing backwards to prove it's correct)

4x3=12 

5x10=50

12+50=62

For this problem, you can guess and check.

If we have 7 bags of 9 apples, it totals 63 apples.

If we have 7 bags of 5 apples, it toals 35 apples.

Each time we replace a bag of 9 apples with 5, it deducts 4 apples from the total.

63-51=12 apple difference from 7 bags of 9 apples from the actual number of apples

12/4=3 bags of red apples

7-3=4 bags of green apples

Therefore, there are 4 bags of green apples and 3 bags of red apples.

(doing it backwards to prove that it's correct)

4x9=36

3x5=15

36+15=51.

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There were 500 more fiction books than non-fiction books in a bookshop. When more books were added to the bookshop, the number of fiction books increased by 1/4, while the number of non-fiction books increased by 2/3. The total number of books became 3145. How many non-fiction books were there after the increase?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Fiction books = 500 + x

Non-fiction books = x

(500 + x) + 1/4(500 + x) + x + 2/3(x) = 3145

x = 864

There were 1440 non-fiction books

you made a simplifying error ;-; ... a + b = 5a - 15b...not 3b....so 16b = 4a, so a/b = 1/4

Case 1: 3 men: 7 C 3 * 6 C 2 = 525

Case 2: 4 men: 7 C 4 * 6 C 1 = 210

Case 3: 5 men: 7 C 5 = 21

525 + 210 + 21 = 756.

is my answer below correct?

Simplifying gives \(x^2 + 7x - 1 = 0\), so the roots are \(x = {-7 \pm \sqrt{7^2-4*1*-1} \over 2}\)\(= {-7 \pm \sqrt{53} \over 2}\). \(({{-7 + \sqrt{53}} \over 2}) ^2 + ({{-7 - \sqrt{53}} \over 2})^2 = 51\)

if you move things around you will find this is a quadratic, and you will get x^2 + 7x - 1 = 0 (i believe) and then you can simply plug it in to the quadratic formula

Easy

Let \(s\) be the number of science fiction, and \(m\) be the number of mystery novels. We can form 2 equations: \(9s + 2m = 15, s + m = 4\). Solving those 2 equations gives \(s = 1, m = 3\)

By similarity, \({30 \over 30 + x} = {y \over y + 15} = {14 \over 22} = {7 \over 11}\), and you want to find \(y\). Using \({ y \over y + 15} = {7 \over 11}\), we get \(11y = 7y + 105\), so \(4y = 105\), so \(BC = y = {105 \over 4}\).

y / 14 = (y + 15) / 22

Guess #1: 3 science fiction novels and 1 mystery novel

Find the number of books:

3 + 1 = 4

Find the total cost in dollars:

(9 × 3) + (2 × 1) = 27 + 2 = 29

The guess is incorrect. Try another guess.

Guess #2: 2 science fiction novels and 2 mystery novels

Find the number of books:

2 + 2 = 4

Find the total cost in dollars:

(9 × 2) + (2 × 2) = 18 + 4 = 22

The guess is incorrect. Try another guess.

Guess #3: 1 science fiction novel and 3 mystery novels

Find the number of books:

1 + 3 = 4

Find the total cost in dollars:

(9 × 1) + (2 × 3) = 9 + 6 = 15

The guess is correct. Uncle Porter bought 1 science fiction novel and 3 mystery novels.

Guest is wrong.

The steps to solving it is like this:

We take random sets of three consecutive integers, add them, and find the factors of them.

First set: 1+2+3=6

Second set: 2+3+4=9

Third Set: 3+4+5=12

We can clearly see that the common factor for every set of three is 3.

I hope this helps.

The domain is (-inf,-3] U [3,inf).

d + f = 15.

The prime number is 2.

1in:3ft (scale)

5in:15ft (mulitiply 5 inches by 3ft)

15x6=90ft (mulitiply 15ft by the total number of sides)

Perimerter is 90ft

W : X = 5 : 9

(5x - 2/5(5x)) / 9x - 84 = 1/2

x = 28

5(28) + 9(28) = 392

If I'm not mistaken, Q*E*D = 66 - 322i

x/10 + x/10 + 6 + 82 = x

x = 110

Ken sold   x/10 + 6 = 17

I didn't use the graph.

The graph was just checking that the algebra was correct.

Slope is just rise over run

Rise is just the difference in y values and run is the difference in x values.  Easy peazy.

(just be careful about the negatives.)

Sorry I had to go. It's not possible to find k, like OrangeJuicy said.