15-344/Homework Assignment 8 - Revision history

Drorbn at 15:39, 23 December 2015

2015-12-23T15:39:04Z

← Older revision		Revision as of 11:39, 23 December 2015
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	Homework Assignment 8 Solution		Homework Assignment 8 Solution

	'''9a) Show by combinatorial argument that <math>\binom{2n}{n} = 2 \binom{n}{2} + n^2 </math>'''		'''9a) Show by combinatorial argument that <math>\binom{2n}{2} = 2 \binom{n}{2} + n^2 </math>'''

	'''A)'''Split <math> 2n </math> into two groups of n. Then choosing 2 out of <math> 2n </math> can be seen as either choosing 2 from group 1 only		'''A)'''Split <math> 2n </math> into two groups of n. Then choosing 2 out of <math> 2n </math> can be seen as either choosing 2 from group 1 only

Octopusheng at 05:54, 16 December 2015

2015-12-16T05:54:09Z

← Older revision		Revision as of 01:54, 16 December 2015
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	, which is <math>\binom{n}{2} </math> or 2 from group 2 only, which is also <math>\binom{n}{2} </math>, or one from each group, which is		, which is <math>\binom{n}{2} </math> or 2 from group 2 only, which is also <math>\binom{n}{2} </math>, or one from each group, which is
	<math> n^2 </math>. Their sum gives the total number of possibilities, which is exactly the right hand side of the equality.		<math> n^2 </math>. Their sum gives the total number of possibilities, which is exactly the right hand side of the equality.

			== Scanned Solution to Homework8 P7,11,15 ==
			<gallery>
			15-344-HW8-Yunheng Chen.jpg\|Page 1
			</gallery>

ChrisKim at 02:03, 16 December 2015

2015-12-16T02:03:55Z

← Older revision		Revision as of 22:03, 15 December 2015
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	Homework Assignment 8 Solution		Homework Assignment 8 Solution

	'''9a)~~'''~~ Show by combinatorial argument that <math>\binom{2n}{n} = 2 \binom{n}{2} + n^2 </math>		'''9a) Show by combinatorial argument that <math>\binom{2n}{n} = 2 \binom{n}{2} + n^2 </math>'''

	'''A)'''Split <math> 2n </math> into two groups of n. Then choosing 2 out of <math> 2n </math> can be seen as either choosing 2 from group 1 only		'''A)'''Split <math> 2n </math> into two groups of n. Then choosing 2 out of <math> 2n </math> can be seen as either choosing 2 from group 1 only

ChrisKim at 01:24, 16 December 2015

2015-12-16T01:24:24Z

← Older revision		Revision as of 21:24, 15 December 2015
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	'''9a)''' Show by combinatorial argument that <math>\binom{2n}{n} = 2 \binom{n}{2} + n^2 </math>		'''9a)''' Show by combinatorial argument that <math>\binom{2n}{n} = 2 \binom{n}{2} + n^2 </math>

	Split <math> 2n </math> into two groups of n. Then choosing 2 out of <math> 2n </math> can be seen as either choosing 2 from group 1 only		'''A)'''Split <math> 2n </math> into two groups of n. Then choosing 2 out of <math> 2n </math> can be seen as either choosing 2 from group 1 only
	, which is <math>\binom{n}{2} </math> or 2 from group 2 only, which is also <math>\binom{n}{2} </math>, or one from each group, which is		, which is <math>\binom{n}{2} </math> or 2 from group 2 only, which is also <math>\binom{n}{2} </math>, or one from each group, which is
	<math> n^2 </math>. Their sum gives the total number of possibilities, which is exactly the right hand side of the equality.		<math> n^2 </math>. Their sum gives the total number of possibilities, which is exactly the right hand side of the equality.

ChrisKim at 01:24, 16 December 2015

2015-12-16T01:24:10Z

← Older revision		Revision as of 21:24, 15 December 2015
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	{{15-344:Dror/Students Divider}}		{{15-344:Dror/Students Divider}}

			Homework Assignment 8 Solution

			'''9a)''' Show by combinatorial argument that <math>\binom{2n}{n} = 2 \binom{n}{2} + n^2 </math>

			Split <math> 2n </math> into two groups of n. Then choosing 2 out of <math> 2n </math> can be seen as either choosing 2 from group 1 only
			, which is <math>\binom{n}{2} </math> or 2 from group 2 only, which is also <math>\binom{n}{2} </math>, or one from each group, which is
			<math> n^2 </math>. Their sum gives the total number of possibilities, which is exactly the right hand side of the equality.

Drorbn at 18:08, 26 November 2015

2015-11-26T18:08:27Z

← Older revision		Revision as of 14:08, 26 November 2015
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	'''Reread''' sections 5.1-5.5 of our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 6.1-6.3, just to get a feel for the future.		'''Reread''' sections 5.1-5.5 of our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 6.1-6.3, just to get a feel for the future.

	'''Solve and submit''' problems 7, 9, 11, and 15 in section 5.5.		'''Solve and submit''' problems 7, 9a, 11, and 15 in section 5.5. (9b was originally also assigned, but it is too difficult).

	{{15-344:Dror/Students Divider}}		{{15-344:Dror/Students Divider}}

Drorbn at 15:54, 26 November 2015

2015-11-26T15:54:44Z

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	This assignment is due <span style="color: blue;">at the tutorials on Thursday November 26</span>. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due <span style="color: blue;">at the tutorials on Thursday November 26</span>. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

	'''Reread''' sections ... of our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections ..., just to get a feel for the future.		'''Reread''' sections 5.1-5.5 of our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 6.1-6.3, just to get a feel for the future.

	'''Solve and submit''' problems 7, 9, 11, and 15 in section 5.5.		'''Solve and submit''' problems 7, 9, 11, and 15 in section 5.5.

Drorbn at 03:59, 21 November 2015

2015-11-21T03:59:30Z

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	{{15-344/Navigation}}		{{15-344/Navigation}}
	{{In Preparation}}
	This assignment is due <span style="color: blue;">at the tutorials on Thursday November 26</span>. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due <span style="color: blue;">at the tutorials on Thursday November 26</span>. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn at 03:58, 21 November 2015

2015-11-21T03:58:27Z

← Older revision		Revision as of 23:58, 20 November 2015
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	'''Reread''' sections ... of our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections ..., just to get a feel for the future.		'''Reread''' sections ... of our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections ..., just to get a feel for the future.

	'''Solve''' problems ~~...~~ in ~~section ...~~, ~~but~~ ~~submit~~ ~~only~~ ~~your~~ ~~solutions of the underlined problems~~.		'''Solve and submit''' problems 7, 9, 11, and 15 in section 5.5.

	{{15-344:Dror/Students Divider}}		{{15-344:Dror/Students Divider}}

Drorbn: Created page with "{{15-344/Navigation}} {{In Preparation}} This assignment is due at the tutorials on Thursday November 26. Here and everywhere, '''neatness co..."

2015-11-19T13:09:06Z

Created page with "{{15-344/Navigation}} {{In Preparation}} This assignment is due <span style="color: blue;">at the tutorials on Thursday November 26</span>. Here and everywhere, '''neatness co..."

New page

{{15-344/Navigation}}
{{In Preparation}}
This assignment is due <span style="color: blue;">at the tutorials on Thursday November 26</span>. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Reread''' sections ... of our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections ..., just to get a feel for the future.

'''Solve''' problems ... in section ..., but submit only your solutions of the underlined problems.

{{15-344:Dror/Students Divider}}