21-127: Concepts of Mathematics
Homework 6 Solutions
5.18 (10 points)
Count the number of 6 card hands, dealt from a standard deck of 52 cards, that have at least one card
in every suit.
Solution:
To have a six card hand with at least one card in every su
Name_
Quiz 2 (50 total points)
09-103 Atoms, Molecules and Chemical Change
11/17/11
Fall 2011
Please print your name on every page of this quiz. Show your work or explanations on
problems or short answer questions as requested in order to receive full cre
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
P ROVING P ROBLEMS NP-C OMPLETE
( L ECTURE 21)
S LIDES FOR 15-453
S PRING 2011
1 / 27
S UMMARY
Complexity Classes: P and NP
Polynomial time reducibility
Satisability Problem (SAT )
CNF, 3CNF Forms
3SAT Proble
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
S PACE C OMPLEXITY
( L ECTURE 22)
S LIDES FOR 15-453
S PRING 2011
1 / 24
S PACE C OMPLEXITY
(Disk) Space the nal frontier!
How much memory do computational problems require?
We characterize problems based on
Design and Implementation of
Speech Recognition Systems
Spring 2011
Bhiksha Raj, Rita Singh
Class 2: Data Capture
24 Jan 2011
Producing the Speech Signal
All sounds are actually pressure waves
The speech producing mechanism is resonant chamber
that prod
Design and Implementation of
Speech Recognition
Speech Recognition Systems
Spring 2011
2011
Class 3: Feature Computation
26 Jan 2011
11756/18799D
1
First Step: Feature Extraction
Speech recognition is a type of pattern recognition problem
Q: Should the
Design and Implementation of
Speech Recognition Systems
Spring 2011
Class 4: Dynamic Time Warping
31 Jan 2011
31 Jan 2011
1
English or German?
The European Commission has just announced that English, and not German, will be the
official language of the Eu
Homework 7
Sidharth Madan
smadan1@andrew.cmu.edu
C
November 3, 2011
A15
A16
The lattice path between two points is determined by which steps are taken to the right.
Thus, the path is determined by the binary n-tuple. This establishes a one-to-one correspo
21-127: Concepts of Mathematics
Homework 6 (due Thursday, October 20)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle task is to keep it simple and succinct. Your solut
Homework 6
Sidharth Madan
smadan1@andrew.cmu.edu
C
October 20, 2011
5.18
5.21
5.31
We can arrange 2n people into pairs of two by picking them successively
n
This can be done in 22 2n2 2 2n2 4 . 2 ways
2
But this leads to a lot of double counting of pairs,