Project Euler/172 - Revision history

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2019-10-09T03:46:22Z

Replacing charlesreid1.com:3000 with git.charlesreid1.com

← Older revision		Revision as of 03:46, 9 October 2019
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	How many 18-digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n?		How many 18-digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n?

	Solution: https://charlesreid1.com~~:3000~~/cs/euler/src/master/scratch/Round7_170-180/Problem172.py		Solution: https://git.charlesreid1.com/cs/euler/src/master/scratch/Round7_170-180/Problem172.py

	==Explanation==		==Explanation==

2019-03-24T20:09:04Z

Final Code

← Older revision		Revision as of 20:09, 24 March 2019
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	===Final Code===		===Final Code===

	Final code: https://charlesreid1.com~~:3000~~/cs/euler/src/master/scratch/Round7_170-180/Problem172.py		Final code: https://git.charlesreid1.com/cs/euler/src/master/scratch/Round7_170-180/Problem172.py

	===Final Answer===		===Final Answer===

2018-02-14T02:04:00Z

Final Answer

← Older revision		Revision as of 02:04, 14 February 2018
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	</math>		</math>


			===Final Code===

			Final code: https://charlesreid1.com:3000/cs/euler/src/master/scratch/Round7_170-180/Problem172.py

	===Final Answer===		===Final Answer===

2018-02-14T02:03:43Z

Problem Statement

← Older revision		Revision as of 02:03, 14 February 2018
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	How many 18-digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n?		How many 18-digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n?

			Solution: https://charlesreid1.com:3000/cs/euler/src/master/scratch/Round7_170-180/Problem172.py

	==Explanation==		==Explanation==

2018-01-17T08:59:51Z

Multiset Permutations

← Older revision		Revision as of 08:59, 17 January 2018
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	Note that multiset permutations are also discussed on the following pages:		Note that multiset permutations are also discussed on the following pages:
	* [[Project Euler/15]]		* [[Project Euler/15]]
			* [[Project Euler/502]]
	* [[AOCP/Multisets]]		* [[AOCP/Multisets]]
	* https://charlesreid1.github.io/shortest-lattice-paths-and-multiset-permutations.html		* https://charlesreid1.github.io/shortest-lattice-paths-and-multiset-permutations.html

2018-01-17T08:42:04Z

Final Answer

2018-01-17T08:40:29Z

Generalizing

← Older revision		Revision as of 08:40, 17 January 2018
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	===Generalizing===		===Generalizing===

	We can generalize the process. Suppose we are forming a number of length <math>N</math> from a number of digits/classes <math>k<math> labeled from <math>0 \dots k-1</math>, and each digit/class can only appear a maximum of <math>m</math> times.		We can generalize the process. Suppose we are forming a number of length <math>N</math> from a number of digits/classes <math>k</math> labeled from <math>0 \dots k-1</math>, and each digit/class can only appear a maximum of <math>m</math> times.

	The number of combinations that can be formed for a given <math>N, k, m</math> is given by the multiset permutation expression above. So the total number of permutations that can be formed is a sum of these multiset permutation expressions, over each possible combination of digits/classes into a number of length <math>N</math>.		The number of combinations that can be formed for a given <math>N, k, m</math> is given by the multiset permutation expression above. So the total number of permutations that can be formed is a sum of these multiset permutation expressions, over each possible combination of digits/classes into a number of length <math>N</math>.

2018-01-08T03:53:41Z

Multiset Permutations

← Older revision		Revision as of 03:53, 8 January 2018
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	===Multiset Permutations===		===Multiset Permutations===

	~~Also see article~~ on lattice paths.		Note that multiset permutations are also discussed on the following pages:
			* [[Project Euler/15]]
			* [[AOCP/Multisets]]
			* https://charlesreid1.github.io/shortest-lattice-paths-and-multiset-permutations.html

	If we are selecting from a group of <math>N_1</math> things of type A, <math>N_2</math> things of type B, and <math>N_3</math> things of type C to form a total of <math>N</math> things, this type of combinatorics problem is called a multiset permutation, and the total number of ways of arranging this set of 3 things is given by:		If we are selecting from a group of <math>N_1</math> things of type A, <math>N_2</math> things of type B, and <math>N_3</math> things of type C to form a total of <math>N</math> things, this type of combinatorics problem is called a multiset permutation, and the total number of ways of arranging this set of 3 things is given by:

2018-01-08T01:49:25Z

Pseudocode

← Older revision		Revision as of 01:49, 8 January 2018
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	<math>		<math>
	N_0 = N - N_1 - N_2 - \dots - N_8 - N_9		N_0 = N - N_1 - N_2 - \dots - N_8 - N_9
			</math>


			===Final Answer===

			Setting the correct parameters should result in the following result:

			<math>
			P = 227485267000992000
	</math>		</math>

2018-01-08T01:48:33Z

Test Case 2

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	<math>		<math>
	\binom{6}{0,3,3} + \binom{6}{1,2,3} + \binom{6}{1,3,2} + \binom{6}{2,1,3} + \binom{6}{2,2,2} + \binom{6}{2,3,1} + \binom{6}{3,0,3} + \binom{6}{3,1,2} + \binom{6}{3,2,1} + \binom{6}{3,3,0}		a = \binom{6}{0,3,3} + \binom{6}{1,2,3} + \binom{6}{1,3,2} + \binom{6}{2,1,3} + \binom{6}{2,2,2} + \binom{6}{2,3,1} + \binom{6}{3,0,3} + \binom{6}{3,1,2} + \binom{6}{3,2,1} + \binom{6}{3,3,0}
	</math>		</math>

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	<math>		<math>
	0 + \binom{5}{0,2,3} + \binom{5}{0,3,2} + \binom{5}{1,1,3} + \binom{5}{1,2,2} + \binom{5}{1,3,1} + \binom{5}{2,0,3} + \binom{5}{2,1,2} + \binom{5}{2,2,1} + \binom{5}{2,3,0}		b = 0 + \binom{5}{0,2,3} + \binom{5}{0,3,2} + \binom{5}{1,1,3} + \binom{5}{1,2,2} + \binom{5}{1,3,1} + \binom{5}{2,0,3} + \binom{5}{2,1,2} + \binom{5}{2,2,1} + \binom{5}{2,3,0}
	</math>		</math>

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	Out[42]: 340.0		Out[42]: 340.0
	</pre>		</pre>

			<math>
			a - b = 340
			</math>

	===Recursion===		===Recursion===