Prove the following two identities both algebraically and by interpr.pdf

•

0 likes•4 views

Prove the following two identities both algebraically and by interpreting their meaning combinatorially. 1. (n chooses k)= (n chooses n-k) 2.(n chooses k) = ((n-1) chooses (k-1)) + ((n-1) choose k)) Solution 1. nCk = n! / (k!.(n-k)!) nC(n-k) = n! / [(n-k)!.k!] Hence algebraically equal. selection of k out of n and rejection of (n-k) objects out of n is the same thing. Hence these are equal. 2. (n-1) C (k-1) + n-1 C k = (n-1)!/[(k-1)!.(n-k)!] + (n-1)!/[k!(n-k-1)!] = (n-1)!/[(k-1)!(n-k-1)] * (1/n-k + 1/k) = (n-1)!/[(k-1)!(n-k-1)] * n/[(n-k).k] = n! / k!.(n-k)! =nCk.

More from aishwaryaenterprises2

Provide an example of a divisible asset. Please be sure to explain how this is a divisible asset. Solution Divisible asset includes all the assets that can be taken by the trustee including house, land, plant, equipment, or any vehicle. These are considered as divisible assets due to the following reasons: During the bankruptcy of a person these assets can be acquired by the trustee. The trustee would become the legal owner of the property and he can sell the property to make payments to the creditors..

Provide an example of a divisible asset. Please be sure to explain h.pdf

aishwaryaenterprises2

provide an example from personal experience or from the literature illustrating how groundwater pumping on drawdown of the water table both locally and at the valley scale may have led to an alteration of downstream flows or to adverse effects on some other component of this system. What properties of the aquifer/surface water system may have made this system more susceptible to such change? Solution Let us understand first the concept of ground water pumping:The water that enters the earth strata gets collected beneath earth to a level that level is called water table , It is measured is metres.The water is infiltrated through the phreatic line.In todays world we are using underground water in excess this has led to deepening of water table level.The aquifier is affected by both locally as construction activities dont let water to infilitritate, Loweing of water is a phenomena occured due to this adverse effect, This also results in increase in TDS of water and also salt water comes to the surface . The properties of aquifier / surface system that make this worse are :Surface water bodies generally gain water and solutes from ground water this phenomena increase the impact of lowering of water table.The excessive use of water in a state leads to water level lowering A vertical section of flow field indicates how potential energy is distributed beneath the water table inthe ground water and surface water. The quantity of ground water discharge to surface water bodies can be determined for a known cross section by multiplying the Hydraulic gradient.The soil properties alos make it more susceptible to drawdown of water table.Perneability is measure of movement of water that also makes a large impact on the drawdown as example sand is more permeable than clay because pore space between sand grains are larger than particles of clay. Transpiration by plants also affects the drawdown condition to increase. Now the interaction between in three basic ways : stream gain water from stream (whic increases drawdown)They both gain and lose but when drawdown is high it impacts the underground water adversely..

provide an example from personal experience or from the literature i.pdf

aishwaryaenterprises2

Provide an appropriate response. The finalists in an essay competition are Lisa (L), Melinda (M), Ben (B), Danny (D), Eric (E), and Joan (J). Consider these finalists to be a population of interest. The possible samples (without replacement) of size two that can be obtained from this population of six finalists are as follows. L,ML,BL,DL,EL,JM,BM,DM,EM,JB,DB,EB,JD,ED,JE,J If a simple random sampling method is used to obtain a sample of two of the finalists, what are the chances of selecting Lisa and Danny? Solution 1/16.

Provide an appropriate response.The finalists in an essay compet.pdf

aishwaryaenterprises2

Prove this identity by making a combinatorial proof, that mean to make a counting argument using photos and/or committees. Solution THE LHS = 0*nC0 + 1*nC1 + 2*nC2 + ... + n*nCn THE RHS = Notice the n is counting the total number of subsets we have from the LHS Furthermore notice that we are also counting all possible subsets. For example nC1 counts all 1 element subsets from a set of n elements. nC2 counts all 2 element subsets from a set of n elements. It is known that for any set with n element that there are 2^n subsets. Why? Well, the most logical way to think about this is saying for each object in the set we can either include it in the subset or not include it. So 2 options for each of n elements is 2 * 2 * 2 ... * 2 (n times) = 2^n. So why do we have 2*n-1 on the RHS? Well as you can see we aren\'t counting the empty set on the LHS because of the 0 * nC0 = 0 Thus, we have shown that the LHS and the RHS count exactly the same thing. This completes the proof..

Prove this identity by making a combinatorial proof, that mean to ma.pdf

aishwaryaenterprises2

Provide a detailed definition of probability value. Solution Consider an experiment where you\'ve measured values in two samples, and the means are different. How sure are you that the population means are different as well? There are two possibilities: The populations have different means. The populations have the same mean, and the difference you observed is a coincidence of random sampling. The P value is a probability, with a value ranging from zero to one. A p-value is a measure of how much evidence we have against the null hypothesis..

Provide a detailed definition of probability value.Solution .pdf

aishwaryaenterprises2

Provide a numerical example of a basic/key financial ratio and explain its organizational implications! (Criteria for Learning Activity evaluation is as follows: promptness and initiative; critical thinking and demonstration of knowledge, listening, attentiveness, community awareness; formatting, structure, writing mechanics; and, APA usage.) Solution There are various financial ratios. I am going to exemplify Return on Equity. A company has the following information: Net operating income = 500,000 Average Shareholder.

Provide a numerical example of a basickey financial ratio and expla.pdf

aishwaryaenterprises2

Prove the following theorem. let sn be a bounded sequence then li.pdf

aishwaryaenterprises2

Prove the identity cosh(x+y)=coshxcoshy+sinhxsinhy. I think I understand why it\'s a positive sign but I don\'t know how to show it algebraically... Solution coshx = (e^x + e^-x)/2, sinhx = (e^x - e^-x)/2, then coshxcoshy + sinhxsinhy = (1/2)(e^x + e^-x)(1/2)(e^y + e^-y) + (1/2)(e^x - e^-x) (1/2)(e^y - e^- y) = (1/4)(e^x + e^-x)(e^y + e^-y) + (1/4)(e^x - e^-x) (e^y - e^-y) = (1/4)[e^(x+y) + e^(x-y) + e^(-x+y) + e^(-x-y)] + (1/4)[e^(x+y) - e^(x-y) - e^(-x+y) + e^(-x-y)] = (1/4)[2e^(x+y)+2e^(-x-y)] = (1/2)[e^(x+y) + e^-(x+y)] = cosh(x+y)..

Prove the identity cosh(x+y)=coshxcoshy+sinhxsinhy. I think I un.pdf

aishwaryaenterprises2

Prove the following theoremsTheorem Log2(11) is irrationallo.pdf

aishwaryaenterprises2

Prove the following inequality by induction on nSolutionLet n.pdf

aishwaryaenterprises2

More from aishwaryaenterprises2 (10)