Recent questions tagged permutation

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Carol Lee's Donuts on North Main Street makes 23 kinds of donuts every morning. Your dad gives you some money and instructions to buy a dozen donuts from their shop. "But...
699
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How many ways can you choose 3 numbers from a Lotto ticket with 6 numbers?
248
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Permutation vs. combination
529
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Calculate, by using an appropriate formula: \(3-1+\frac{1}{3}-\ldots \ldots \ldots \ldots \ldots+\frac{1}{243}\)
522
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Consider the numbers \(-3 ;-\frac{7}{4} ; 2,25\) and choose the correct combinations that apply.i: The numbers are integers and rational numbersii: The numbers are natura...
294
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If you randomly select four people from a group of 100 people, what is the probability that at least one of their names is Sipho?
569
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If you randomly select three people from a group of 100 people, what is the probability that one of their names is Nomsa and the other two are Thabo?
274
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If you randomly select two people from a group of 100 people, what is the probability that both of their names are Thando?
224
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If you randomly select one person from a group of 100 people, what is the probability that their name is Zandile?
325
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Can anyone ask a question on Acalytica Q&A, or are there certain criteria that must be met?
310
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Complete the following number sequence.\[3 ; 9 ; 27 ; 81 \text {; .. }\]
265
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Complete the following number sequence.\[1 ;-3 ; 9 ;-27 ;\]
433
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17 scientists regularly discuss three particular topics. Any two of them only discuss one of the three topics with each other. Prove that there is a group of three scient...
280
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Prove that among any six people there are always either three who know each other or three who are unknown to each other.
355
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All the houses on one side of a certain street are to be painted either yellow or red. In how many ways can this be done if there are \(n\) houses and there may not be tw...
247
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King Arthur chooses three of the 25 knights sitting around his table to fight a fearsome dragon. How many possible choices are there, if no two of the chosen knights shou...
146
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Prove that the number of possibilities to arrange \(n\) (distinguishable) objects in a row (so-called permutations) is\[n !=n \cdot(n-1) \cdot \ldots \cdot 3 \cdot 2 \cdo...
236
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Given a finite sequence \(x_{1,1}, x_{2,1}, \ldots, x_{n, 1}\) of integers ( \(n \geq 2\) ), not all equal, define the sequences \(x_{1, k}, \ldots, x_{n, k}\) by\[x_{i, ...
176
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Prove that\[\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}<\frac{2}{3}\]for all \(n \geq 2\)
157
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Prove that\[\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}<\frac{2}{3}\]for all \(n \geq 2\)
172
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Given that \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1334}+\frac{1}{1335}=\frac{p}{q}\), where \(p\) and \(q\) are coprime natural numbers, prove that \(p\)...
386
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The sum of coefficient of first, second, and third terms in \(\left(x^{2}+\frac{1}{x}\right)^{\mathbf{n}}\) is 46 then coefficient of term that does not contain \(\mathrm...
513
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Suppose that \(x_{1}, x_{2}, \ldots, x_{n}\) are positive real numbers. Prove that\[\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{3}}+\cdots+\frac{x_{n-1}}{x_{n}}+\frac{x_{n}}{x_{1...
334
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Define a sequence \(a_{1}, a_{2}, \ldots\) by \(a_{1}=\frac{5}{2}\) and \(a_{n+1}=a_{n}^{2}-2\) for \(n \geq 1\). Give an explicit formula for \(a_{n}\) and prove it.
244
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Each of the \(3 n\) members of a parliament slapped one of his/her colleagues. Prove that among them it is possible to choose a committee consisting of \(n\) members none...
707
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Out of 5 mathematicians and 7 engineers, a committee consisting of 2 mathematicians and 3 engineers is to be formed. In how many ways can this be done if(a) any mathemati...
341
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Consider arranging 3 letters: A, B, C. How many ways can this be done?
347
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For our clothes problem above, say we found 3 caps that we could wear with our 2 t-shirts and 4 pairs of jeans. How many different combinations could we choose from now?
464
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Say the only clean clothes you've got are 2 t-shirts and 4 pairs of jeans. How many different combinations can you choose?
596
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Our club has 25 members, and wishes to pick a president, secretary, and treasurer. In how many ways can we choose the officers, if individual members are allowed to hold ...
441
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How many ways are there to divide 12 people into a group of 3, a group of 4, and a group of 5, if Henry has to be in the group of 4?
749
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In how many ways can we seat 6 people at a round table if John and Sam insist on sitting next to each other? (Two seatings are equivalent if one is a rotation of the othe...
352
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Your math club has 20 members. In how many ways can it select a president, a vice-president, and a treasurer if no member can hold more than one office?
539
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In how many ways can we form an international commission if we must choose one European country from among 6 European countries, one Asian country from among 4, one North...
280
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In a group of 30 high school students, 8 take French, 12 take Spanish and 3 take both languages. How many students of the group take neither French nor Spanish?
275
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How many ways can a president, a vice president and a 3 person advisory committee be assigned from 30 people?
516
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In how many ways can 100 students be assigned 40 into dorm \(A, 35\) into dorm \(B\) and 25 into dorm C?
752
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How many ways can a company select 4 candidates to interview from a short list of 12 engineers?
488
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Determine the number of ways in which a software professional can choose 4 of 25 laptops to test a newly designed application.
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In how many different ways can the director of a research laboratory choose 2 chemists from among 7 applicants and 3 physicists from among 9 applicants?
850
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In how many different ways can 3 of 18 automotive engineers be chosen for a team to develop a new ceramic diesel engine.
643
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In how many different ways can a union local with a membership of 25 choose a vice president and a president?
460
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If \(k\) sets of data consist, respectively, of \(n_{1}, n_{2}, \ldots, n_{k}\) observations and have the means \(\bar{x}_{1}, \bar{x}_{2}, \ldots, \bar{x}_{k}\), then th...
260
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Show that\[\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)=0\]for any set of observations \(x_{1}, x_{2}, \ldots, x_{n}\).
314
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Prove that \(1+3+5+\cdots+2 n-1=n^{2}, \forall n \in N\)
362
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In examination a junior high school student may choose to solve 8 put of 12 word problems. In how many ways can the student choose the word problems?
659
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Evaluate \(\left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{l}n \\ 2\end{array}\right)+\left(\begin{array}{l}n \\ 4\end{array}\right)+\cdots+\left(\begi...
667
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Evaluate \(\left(\begin{array}{l}n \\ 0\end{array}\right)-\left(\begin{array}{l}n \\ 1\end{array}\right)+\cdots+(-1)^{n}\left(\begin{array}{l}n \\ n\end{array}\right)\)
588
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Evaluate \(\left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{l}n \\ 1\end{array}\right)+\cdots+\left(\begin{array}{l}n \\ n\end{array}\right)\)
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Find the common ratio of the geometric sequence \(1,3,9,27, \ldots\) Then express each sequence in the form \(a_{n}=a_{1} r^{n-1}\) and find the eighth term of the sequen...
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