Recent questions in Counting Principles

High school probabilityAnswered question

v1kibc2 2023-03-14

1 octopus has 8 legs. How many legs does 3 octopuses have?

A) 16

B 24

C) 32

D) 14

A) 16

B 24

C) 32

D) 14

High school probabilityAnswered question

funnyantyLEy 2022-12-01

What are the applications of Archimedes' principle ? AIt is used in designing of ships and submarines. BIt is used in lactometers to determine the purity of milk. CIt is used in hydrometers to determine density of fluids. DIt is used in hydraulic lifts.

High school probabilityAnswered question

BenguelaktR 2022-11-23

Determine the number of different ways that the letters of "success" can be arranged

High school probabilityAnswered question

Jenny Schroeder 2022-11-15

How many license plates can be made consisting of 2 letters followed by 3 digits (using the fundamental counting principle to solve)? What I know:

1. 26 letters in alphabet, so that means $2\times 26$

2. 10 digits possible (0-9), so that means $3\times 10$

3. FC principle says given m and n options gets you $m\times n$ varieties...

... However, the answer key says "676,000" when I got 1560...

1. 26 letters in alphabet, so that means $2\times 26$

2. 10 digits possible (0-9), so that means $3\times 10$

3. FC principle says given m and n options gets you $m\times n$ varieties...

... However, the answer key says "676,000" when I got 1560...

High school probabilityAnswered question

pighead73283r 2022-11-05

Different 4 math books , 6 different physics books , 2 different chemistry books are to be arranged on a shelf. how many arrangement is possible if the books in particular subject must stand together ?

High school probabilityAnswered question

kaltEvallwsr 2022-11-02

Of a group of 50 students, 20 are freshmen, 10 are sophomores, 15 are juniors, and 5 are seniors. 5 members must be chosen. How many different committees can there be if there must be exactly one senior and exactly two freshmen on the committee?

High school probabilityAnswered question

taumulurtulkyoy 2022-10-20

How do you use the Counting Principle to find the probability of choosing the 5 winning lottery numbers when the numbers are chosen at random from 0 to 9?

High school probabilityAnswered question

Christopher Saunders 2022-10-18

A bag contains 8 white marbles and 7 blue marbles. Find the probability of selecting 2 white marbles and 3 blue marbles: (hint: use combinations and fundamental counting principle)

It doesn't say it, but I'm sure there are no replacements. I know it's a combination, but I don't know how 3C2 (enteblack it equals 3) would be the answer because just "3" doesn't sound right.

It doesn't say it, but I'm sure there are no replacements. I know it's a combination, but I don't know how 3C2 (enteblack it equals 3) would be the answer because just "3" doesn't sound right.

High school probabilityAnswered question

Bodonimhk 2022-10-13

A piece of string $25\frac{1}{2}$ inches long will be cut into $\frac{3}{4}$-inch pieces. How many pieces will there be?

High school probabilityAnswered question

tamnicufl 2022-09-27

There are $\frac{m}{gcd(m,x)}$ distinct elements in the set $\{ax\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}m):a\in \{0,...,m-1\}\}$

I have only known these by using a computer to generate the number of distinct elements. But I am not sure how to prove this conjecture.

And is there any way that we can connect this problem to Euler's phi function so that we can simply use properties of $\varphi $ function to prove it?

And can we also use some counting principle here to give an exact answer?

I have only known these by using a computer to generate the number of distinct elements. But I am not sure how to prove this conjecture.

And is there any way that we can connect this problem to Euler's phi function so that we can simply use properties of $\varphi $ function to prove it?

And can we also use some counting principle here to give an exact answer?

High school probabilityAnswered question

Colten Andrade 2022-09-27

Find the number of Sylow 2-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo 3). I think it will be 1. But I failed to prove it using the counting principle. It has 4 sylow 3-subgroups.

High school probabilityAnswered question

trkalo84 2022-09-22

How many numbers $n<100$ are not divisible by a square of any integer greater than 1?

Working through the above counting problem. I got 48 using the Inclusion-Exclusion Principle, do you agree?

Working through the above counting problem. I got 48 using the Inclusion-Exclusion Principle, do you agree?

High school probabilityAnswered question

madeeha1d8 2022-09-20

In how many ways can 7 people be seated in a row o chairs if Jane and Joe must sit next to each other?

High school probabilityAnswered question

maredilunavy 2022-09-15

The next three batters on a baseball team have hit percentages of 0.325, 0.250, and 0.275, respectively. What is the probability that the first and third batters will both get a hit, while the second batter does not?

High school probabilityAnswered question

Genesis Gibbs 2022-09-06

I tried the Fundamental Counting principle approach: There are 26 black cards to chose from for the first card. That leaves $26\times 25\times 24\times 23$ left for the other four cards. So $\frac{26\times 26\times 25\times 24\times 23}{5!}$ (because we can arrange 5 cards in $5!$ ways). I get 77740. Apparently the answer is: 454480.

I'd appreciate help clearing up my misunderstanding. I'd also like to see the combinations approach. Thanks!

I'd appreciate help clearing up my misunderstanding. I'd also like to see the combinations approach. Thanks!

High school probabilityOpen question

musicintimeln 2022-08-14

I was doing a seemingly trivial question, and I though it was a simple application of the counting theorem but it turns out it doesn't work. Here's the question

From a deck of 52 cards, how many ways are there to arrange a hand of 5 cards such that all 4 kings are in the hand (order doesn't matter) (the last card can be any non-king)

Now here's my thought process as an application of the counting principle:

$\frac{4\times 3\times 2\times 1\times 48}{5!}$

As we have 4! ways of placing the kings and then the last card can be from 48 other cards. Then we divide by 5! to remove the order. Unfortunately, this produces a non-integer so I was very sad indeed. However, it logically seems like it should work as it follows what I think is valid logic. Could someone explain how to get the correct answer (48) and also more importantly, why my logic was incorrect?

From a deck of 52 cards, how many ways are there to arrange a hand of 5 cards such that all 4 kings are in the hand (order doesn't matter) (the last card can be any non-king)

Now here's my thought process as an application of the counting principle:

$\frac{4\times 3\times 2\times 1\times 48}{5!}$

As we have 4! ways of placing the kings and then the last card can be from 48 other cards. Then we divide by 5! to remove the order. Unfortunately, this produces a non-integer so I was very sad indeed. However, it logically seems like it should work as it follows what I think is valid logic. Could someone explain how to get the correct answer (48) and also more importantly, why my logic was incorrect?

High school probabilityAnswered question

magda8471 2022-08-12

In a computer game, when choosing a character there are 5 choices for hair style, 3 choices for eye color, and 7 choices for outfits. How many different characters are possible if you were to choose 1 hair style, 1 eye color, and 1 outfit?

High school probabilityAnswered question

vangstosiis 2022-07-22

Let X be the set of 10-digit numbers that do not contain all digits 0 through 9 in their decimal representation and do not begin with 0. How many elements does the set X have?

I think this should be done by counting principle. So I have 10 spots, and 10 options for each spot. However, the first number can't be 0, so the first spot has 9 options.

9 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 9

I think the 10th spot only have 9 options since I can't repeat one number (the representation doesn't have all the digits).

I think this should be done by counting principle. So I have 10 spots, and 10 options for each spot. However, the first number can't be 0, so the first spot has 9 options.

9 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 9

I think the 10th spot only have 9 options since I can't repeat one number (the representation doesn't have all the digits).

High school probabilityAnswered question

asigurato7 2022-07-20

How do you use the Counting Principle to find the probability of rolling a 4 on each of 4 number cubes?

High school probabilityAnswered question

Graham Beasley 2022-07-19

There are two fundamental principles of counting; Fundamental principle of addition and fundamental principle of multiplication.

I often got confused applying them. I know that if there are two jobs, say $m$ and $n$, such that they can be performed independently in $m$ and $n$ ways respectively, then either of the two jobs can be performed in $m+n$ ways and when two jobs are performed in succession, they can be performed in $m\times n$ ways.

My question is how to identify whether jobs are independent or in succession?

Is there any simple way to identify this? Are there any keywords?

I often got confused applying them. I know that if there are two jobs, say $m$ and $n$, such that they can be performed independently in $m$ and $n$ ways respectively, then either of the two jobs can be performed in $m+n$ ways and when two jobs are performed in succession, they can be performed in $m\times n$ ways.

My question is how to identify whether jobs are independent or in succession?

Is there any simple way to identify this? Are there any keywords?

Statistics and probability studies always start during your high school studies, which is why counting principle problems and solutions is one of those things where you have to start. If you find it hard to master these concepts, take a look at our questions and answers that have been presented below. These will help you with the counting principles and show helpful samples. The use of logic is paramount in this regard as you must use combinatorics among other things. Do not forget to look at the fundamental counting and its use in the calculations for daily life estimation challenges.