Multinomial Theorem Examples
Multinomial theorem examples,Multinomial Logp1)Multinomial theorem means nothing but how to write down a power of a sum in conditions of powers of the terms in that sum.p2)Multinomial theorem is nothing but rule of a sum in term of rules of the addends. p3)Multinomial theorem is also called a polynomial theorem.
This multinomial is the simplification of the binomial theorem to polynomials.pMultinomial Theorem Examples:p For example, (a + b + c) 3= a3 + b3 + c3 + 3a2b + 3a2c + 3b2a + 3b2c + 3c2a + 3c2b + 6abc.p Now we calculated each coefficient by first expanding (a + b + c) 2 = a2 + b2 + c2 + 2ab + 2bc + 2ac, then self-multiplying it again to get (a + b + c) 3.p On the supplementary this is sluggish process, and can be avoided by means of the multinomial theorem. It solves this method by giving us the closed form for any coefficient we might want.p It is possible to the multinomial coefficients from the conditions by using the multinomial coefficient formula.p Multinomial theorem is nothing but expression sum in term of terms of the addends.
Multinomial theorem is also known as polynomial theorem. This multinomial is the simplification of the binomial theorem to polynomials.p A multinomial allocation is the option sharing of the outcome from a multinomial experiment. The multinomial method defines the chance of any result from a multinomial experiment.pExamples: 4x3y + z. This is the example of multinomial.brMultinomial Log:p Multinomial log is used when the dependent variable inquiry is supposed and consists of more than two categories.p For example multinomial register it would be suitable when annoying to determine what factors predict which major college students choose.p Multinomial log decay is appropriate in cases where the response is not ordinal in nature as in ordered log it.p1. The coefficient of x4 in [(x/2)- (3/x‚)]10 pSolution:br(r+1) th term of the expansion is given bybr10Cr (x/2) 10-r(-3/x‚) rp= 10Cr (x) 10-r(1/ x‚)r(1/2) 10-r(-3) rp= 10Cr(-3) r(1/2) 10-r(x) 10-3rpIn this term (x) 10-3r is equal to x4br? 10-3r = 4 or r = 2pSo coefficient =br10Cr(-3) r(1/2) 10-rp10C2(-3) 2(1/2) 10-2p= [10*9/2]*9*(1/28)p= (45*9)/256br= 405/256pLet n be a positive integer. If the coefficient of 2nd, 3rd, and 4th terms in the expansion of (1 + x)n are in A.P. then the value of n is pSolution:pThe question requires concepts from binomial theorem and concept from arithmetic progression.pCoefficients of 2nd, 3rd, and 4th terms in the expansion of (1 + x)n arebrnC1, nC2, nC3pAccording to arithmetic progression properties, three terms can be taken as a-d, a, and a+d.
Hence sum of first and third terms = 2*second termp= 2(nC2) = nC1 + nC3p= 2 n(n-1)/2 = n + [n(n-1)(n-2)/6]p= n-1 = 1 +[(n‚ - 3n+2)/6]p= n‚ - 3n+2 = 6n-12br= n‚ - 9n+14 = 0br= (n-2)(n-7) = 0pSince 4th term will be there only when n2, n is equal to 7.br
This multinomial is the simplification of the binomial theorem to polynomials.pMultinomial Theorem Examples:p For example, (a + b + c) 3= a3 + b3 + c3 + 3a2b + 3a2c + 3b2a + 3b2c + 3c2a + 3c2b + 6abc.p Now we calculated each coefficient by first expanding (a + b + c) 2 = a2 + b2 + c2 + 2ab + 2bc + 2ac, then self-multiplying it again to get (a + b + c) 3.p On the supplementary this is sluggish process, and can be avoided by means of the multinomial theorem. It solves this method by giving us the closed form for any coefficient we might want.p It is possible to the multinomial coefficients from the conditions by using the multinomial coefficient formula.p Multinomial theorem is nothing but expression sum in term of terms of the addends.
Multinomial theorem is also known as polynomial theorem. This multinomial is the simplification of the binomial theorem to polynomials.p A multinomial allocation is the option sharing of the outcome from a multinomial experiment. The multinomial method defines the chance of any result from a multinomial experiment.pExamples: 4x3y + z. This is the example of multinomial.brMultinomial Log:p Multinomial log is used when the dependent variable inquiry is supposed and consists of more than two categories.p For example multinomial register it would be suitable when annoying to determine what factors predict which major college students choose.p Multinomial log decay is appropriate in cases where the response is not ordinal in nature as in ordered log it.p1. The coefficient of x4 in [(x/2)- (3/x‚)]10 pSolution:br(r+1) th term of the expansion is given bybr10Cr (x/2) 10-r(-3/x‚) rp= 10Cr (x) 10-r(1/ x‚)r(1/2) 10-r(-3) rp= 10Cr(-3) r(1/2) 10-r(x) 10-3rpIn this term (x) 10-3r is equal to x4br? 10-3r = 4 or r = 2pSo coefficient =br10Cr(-3) r(1/2) 10-rp10C2(-3) 2(1/2) 10-2p= [10*9/2]*9*(1/28)p= (45*9)/256br= 405/256pLet n be a positive integer. If the coefficient of 2nd, 3rd, and 4th terms in the expansion of (1 + x)n are in A.P. then the value of n is pSolution:pThe question requires concepts from binomial theorem and concept from arithmetic progression.pCoefficients of 2nd, 3rd, and 4th terms in the expansion of (1 + x)n arebrnC1, nC2, nC3pAccording to arithmetic progression properties, three terms can be taken as a-d, a, and a+d.
Hence sum of first and third terms = 2*second termp= 2(nC2) = nC1 + nC3p= 2 n(n-1)/2 = n + [n(n-1)(n-2)/6]p= n-1 = 1 +[(n‚ - 3n+2)/6]p= n‚ - 3n+2 = 6n-12br= n‚ - 9n+14 = 0br= (n-2)(n-7) = 0pSince 4th term will be there only when n2, n is equal to 7.br
