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Ksju [112]
3 years ago
8

Pigeon Hole Principle :

Mathematics
1 answer:
yaroslaw [1]3 years ago
5 0

Answer:

A proof can be as follows:

Step-by-step explanation:

Let S=\{a_{1},a_{2},...,a_{n},a_{n+1}\} be a set of n+1 integers. By the division algorithm the possible remainders when we divide by n are 0,1,2,....,n-1. Then, each integer a_{i}\in S can be written as:

a_{i}=np_{i}+r_{i},\,\,0\leq r_{i}

Observe that the set of remainders \{r_{1},r_{2},...,r_{n+1}\} has n+1 elements and each element has n possible values. By the Pigenhole principle at least two remainders have the same value. Suppose that this two elements are a_{i}, a_{j}. Then,

\begin{array}{c}a_{i}=np_{i}+r\\a_{j}=np_{j}+r\end{array}

Where r_{i}=r_{j}=r. Then,  a_{i}-a_{j}=np_{i}-np_{i}=n(p_{i}-p_{j}). Then we have that n divides a_{i}-a_{j}.

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A water container has 19.5 litres of water in it. A cup holds 210 ml of water. At most 92 cups can be filled completely from the
coldgirl [10]

Step-by-step explanation:

Given

A container has 19.5 liters of water

A cup can hold 210 ml of water

1 L is equivalent to 1000 ml

So, 19.5 L is equivalent to 19.5\times 1000=19,500\ ml

So, the number of requires to hold this much water

\Rightarrow n=\dfrac{19,500}{210}\\\\\Rightarrow n=92.85\ \text{cups}

92.85 implies 92 full cups and 1 cup which is partially filled. Hence, almost 92 cups can be filled completely.

5 0
2 years ago
Use the image to match the arc/angle measure.
IrinaK [193]

Answer:

The following measurements are:

m\angle{STR}=23^\circ (Option #4)

m{QT}=142^\circ (Option #7)

mST=134^\circ (Option #5)

mRQ=38^\circ (Option #2)

Step-by-step explanation:

To begin, we can find the measure of \angle{STR} by applying the inscribed angle theorem: an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle.

Since the intercepted arc (RS) is 46 degrees, we have:

46=2\theta\\23=\theta

Next, we can find the measure of arc QT using the same theorem. So,

QT=2(71)\\QT=142

Notice that the chord RT is actually a diameter. From the theorem about the inscribed angle including a diameter, we know that the intercepted arc will have a measure of 180^\circ. Since the arc ST is part of the arc RST, and we know RS is 46^\circ, we can set up and solve this equation:

RST = RS + ST\\180 = 46 + ST\\134 = ST

We can use the same idea to find RQ. We know that RQT is 180^\circ and QT is 142^\circ, so:

RQT = RQ + QT\\180 = RQ + 142\\38 = RQ

7 0
2 years ago
Which of the following is NOT equivalent to 40/100?
frozen [14]

Answer:

Option C

Step-by-step explanation:

40/10 simplifies to 4.

40/100 simplifies to 0.40.

\frac{40/10}{10/10} =\frac{4}{1} =4

\frac{40}{100} =0.40

Therefore:

\frac{40}{10}\neq  \frac{40}{100}

Option C should the the correct answer.

8 0
3 years ago
Can someone help me find the side of a 30 60 90 triangle​
Anastasy [175]

Answer:

y = 7.5

x = 7.5√3

Step-by-step explanation:

sin(30) = 1/2

cos(30) = √3/2

Above is rule

==>

y = 7.5

x = 7.5√3

4 0
2 years ago
Jackson paints every face of the cube what is the total surface area in square centimeters of the cube Jackson paints?
GREYUIT [131]
Find the area of each face and add them all together, or since a cube has 6 faces, you can multiply the area of one face by 6.
7 0
3 years ago
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