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2 years ago

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A science fair awards prizes to the first, second and third place winners. There are 48 people entered in the science fair. How

many ways can the winners be selexted EXPRESSION

1 answer:

vichka [17]2 years ago

5 0

Answer:

Step-by-step explanation:

103,776 ways

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Problems 2.21, 2.22, 2.23

RoseWind [281]

Statements can be proved by contrapositive, contradiction or by induction.

<em>2.21 and 2.23 are proved by contrapositive</em>
<em>2.22 is proved by induction</em>

<u />

<u />

<u>2.21: If </u> $n^3$ <u> is even, then n is even (By contrapositive)</u>

The contrapositive of the above statement is that:

<em>If n is odd, then </em> $n^3$ <em> is odd</em>

Represent the value of n as:

$n = 2k + 1$ , where $k \ge 0$

Take the cube of both sides

$n^3 = (2k + 1)^3$

Expand

$n^3 = 8k^3 + 6k^2 + 6k + 1$

Group

$n^3 =[ 8k^3 + 6k^2 + 6k] + 1$

Factor out 2

$n^3 =2[4k^3 + 3k^2 + 3k] + 1$

Assume w is an integer; where:

$w =4k^3 + 3k^2 + 3k$

So, we have:

$n^3 =2w + 1$

The constant term (i.e. 1) means that $n^3$ is odd.

Hence, the statement has been proved by contrapositive.

<em>i.e. If n is odd, then </em> $n^3$ <em> is odd</em>

<u />

<u>2.22 </u> $3n + 4$ <u> is even, if and only if n is even</u>

We have: $3n + 4$ <u />

<u />

Assume that: $n = 2k + 2$ for $k \ge 0$

So, we have:

$3n + 4 = 3(2k + 2) + 4$

Open bracket

$3n + 4 = 6k + 6 + 4$

$3n + 4 = 6k + 10$

Factorize

$3n + 4 = 2(3k + 5)$

The factor of 2 means that $3n + 4$ is even.

<em>Hence, </em> $3n + 4$ <em> is even, if and only if n is even </em>

<em />

<u />

<u>2.22: </u> $s \ne -1$ <u> and </u> $t \ne -1$ <u>, then </u> $s + t + st \ne -1$ <u />

To do this, we prove by contrapositive.

The contrapositive of the above statement is:

If $s = -1$ and $t=-1$ , then $s + t + st = -1$

We have:

$s + t + st = -1$

Substitute the values of s and t in: $s + t + st = -1$

$-1 -1 -1 \times -1 = -1$

$-1 -1 + 1 = -1$

$-1 = -1$

Hence, by contrapositive:

If $s = -1$ and $t=-1$ , then $s + t + st = -1$

Read more about proofs at:

brainly.com/question/19643658

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3 years ago

Josephine wants to divide the shed into two compartments. One compartment will have a volume of 100ft^3. What is the volume of t

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I believe using the equation v=lwh the answer would be 420ft^3.

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PLSSS HELPPPPP MEEEE III WILLL GIVVVEEE BRAINLIESTTTT!!!!!!

Artyom0805 [142]

They are equal, 8 x 7 = 56 and 7 x 7 = 49

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3 years ago

In which number does the 5 represent 5x 10,000?

lyudmila [28]

It would be 50,000, because 5*10,000=50,000.

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2 years ago

In the equation Q – 12 = 89, what is the value of Q

pav-90 [236]

The answer is 101 because you add 89 and 12

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3 years ago

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