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August 17th, 2018, 10:52 PM   #1
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Binomial Expansion - The notation

Q1: nCr+1 - nCr-1 = n+1Cr-1 - n+1Cr


Q2: nC3 = 5nC1


Q3: nC1+ nC2 = 3n


I would like to know how to solve these questions with detail steps, thank you!!

My answer on Q2 is 1 and Q3 is 4 but the model answer are 7 and 5 respectively.
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August 18th, 2018, 05:43 AM   #2
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(1) Is the equation typed correctly? It reduces eventually to n = 2r - 1, where r > 1.

(2) n(n - 1)(n - 2)/6 = 5n, so n((n - 1)(n - 2) - 30) = 0. Hence n = 0 or 7.

(3) 3n = nC1 + nC2 = n + n(n - 1)/2, so 0 = n(2 + n - 1 - 6). Hence n = 0 or 5.
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August 18th, 2018, 06:11 AM   #3
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Question 1 should be

Prove LHS=RHS
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August 18th, 2018, 12:36 PM   #4
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There must be an error in question (1), as the equation given isn't true in general.

Maybe its right-hand side should have had n+1Cr+1 instead of n+1Cr-1.

It then follows from nCr-1 + nCr = n+1Cr and nCr + nCr+1 = n+1Cr+1.
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August 18th, 2018, 05:59 PM   #5
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Is it possible that you can prove question 1 has an error in detail steps, cause this question is from my textbook so I want to know what is wrong, thank you.
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August 18th, 2018, 06:13 PM   #6
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(1) In its original form, the equation doesn't hold for n = 4 and r = 2, say, so if the third term is n+1Cr-1 in the textbook, the textbook is in error.
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August 18th, 2018, 06:23 PM   #7
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Would you prove that

nC(r+1) - nC(r-1) = (n+1)C(r+1) - (n+1)Cr


Later I contact the publisher to see if there are any typo problem in this question, thanks a lot
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August 18th, 2018, 08:55 PM   #8
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Quote:
Originally Posted by hy2000 View Post
Would you prove that

nC(r+1) - nC(r-1) = (n+1)C(r+1) - (n+1)Cr


Later I contact the publisher to see if there are any typo problem in this question, thanks a lot
Can you at least give it a try yourself and show us what you've done?

-Dan
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August 19th, 2018, 02:17 AM   #9
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Quote:
Originally Posted by hy2000 View Post
Would you prove that

nC(r+1) - nC(r-1) = (n+1)C(r+1) - (n+1)Cr
I've already given two equations that are equivalent to Pascal's rule (also known as Pascal's identity), and which you can use without proof (but several proofs are given in the linked article). If you subtract one of the equations from the other, thus eliminating the nCr terms, the resulting equation can be rearranged to give the equation quoted above that your textbook probably intended.
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Last edited by skipjack; August 19th, 2018 at 07:53 PM.
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August 19th, 2018, 03:15 PM   #10
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Sure I will try it myself, if I have any question I will ask you
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