Algebra Pre-Algebra and Basic Algebra Math Forum

January 30th, 2019, 02:17 AM   #1
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Binomial theorem

How can I prove the equation by using binomial theorem? Thanks.
Attached Images 58D15740-EA8B-4D5C-9122-A77B4CEF7AFC.jpg (17.3 KB, 3 views) January 30th, 2019, 08:16 AM #2 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 552 One way to start is to think $\displaystyle 2^{(n-1)} = (1 + 1)^{(n-1)} = \left (\sum_{j=0}^{n-1} \dbinom{n-1}{j} * 1^{(n-1-j)} * 1^j \right ) = \sum_{j=0}^{n-1} \dbinom{n-1}{j}.$ Now if n is even, I play around. If n = 2 $\dbinom{2-1}{0} + \dbinom{2-1}{1} = \dbinom{1}{0} + \dbinom{1}{1} = \dbinom{2 \div 2}{0} + \dbinom{2 \div 2}{1}.$ What if n = 4? $\dbinom{3}{0} + \dbinom{3}{1} + \dbinom{3}{2} + \dbinom{3}{3} = 1 + 3 + 3 + 1 = 8 =$ $1 + 6 + 1 = \dbinom{4}{0} + \dfrac{4 * 3}{2} + \dbinom{4}{4} = \dbinom{4}{0} + \dbinom{4}{2} + \dbinom{4}{4}.$ Hmm. Maybe $m \in \mathbb Z^+ \text { and } n = 2m \implies \displaystyle \sum_{j=0}^{n-1} \dbinom{n-1}{j} = \sum_{i=0}^{m}\dbinom{n}{2i}.$ Can you prove that? Thanks from justusphung and topsquark January 30th, 2019, 07:11 PM   #3
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Thank you for your hints. My prove is as follow
Attached Images 0ACC3174-C905-40DF-A26E-D46A23D648BC.jpg (20.9 KB, 3 views) January 31st, 2019, 01:39 AM #4 Senior Member   Joined: Aug 2012 Posts: 2,424 Thanks: 759 $\dbinom{n}{k} =$ the number of subsets of size k in a set of size n. Since the number of even and odd subsets of a finite set are equal, we're done. There are $2^n$ subsets, and the expression we're supposed to calculate just counts the number of even subsets, which is therefore $2^{n-1}$. https://math.stackexchange.com/quest...-are-odd-sized Last edited by Maschke; January 31st, 2019 at 01:51 AM. Tags binomial, theorem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post beesee Probability and Statistics 5 September 18th, 2015 02:38 PM jbergin Probability and Statistics 1 December 15th, 2014 10:38 PM Keroro Probability and Statistics 4 June 12th, 2012 04:43 AM mikeportnoy Probability and Statistics 2 March 10th, 2009 07:29 AM John G Applied Math 1 January 12th, 2009 02:40 PM

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