SOLUTION: 37. In this exercise we study the connection between sets (from Chapter 7) and combinations (from Chapter 8). a. Given a set with n elements, what is the number of subsets of siz

Question 250032: 37. In this exercise we study the connection between sets (from Chapter 7) and combinations
(from Chapter 8).
a. Given a set with n elements, what is the number of subsets of size 0? of size 1? of
size 2? of size n?
b. Using your answer from part a, give an expression for the total number of subsets of
a set with n elements.
c. Using your answer from part b and a result from Chapter 7, explain why the following
equation must be true: (n) + (n) + (n) + ... + (n) =2n
(0) (1) (2) (n)
d. Verify the equation in part c for n=4 and n=5
e. Explain what the equation in part c tells you about Pascal�s triangle
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
a. Given a set with n elements, what is the number of subsets
of size 0?---------nC0 = 1
of size 1? of------nC1 = n
size 2?------------nC2= (n(n-1))/2
of size n?---------nCn = 1
----------------------------------------------
b. Using your answer from part a, give an expression for the total number of subsets of a set with n elements.--------------
Total number of subsets = nC0 + nC1 + nC2 + ... + nCn = 2^n
----------------------------------------
c. Using your answer from part b and a result from Chapter 7, explain why the following
equation must be true: nC0 + nC1 + nC2 + ... + nCn = 2^n
Ans: Every element in the set of "n" is either chosen or not
chosen in each subset. So the number of possible subsets
is 2^n (a choice with 2 possible outcomes made n times)
------------------------------------------------------------------
d. Verify the equation in part c for n=4 and n=5
4C0 + 4C1 + 4C2 + 4C3 + 4C4
= 1 + 4 + 6 + 4 + 1
= 16
= 2^4
---------------------------------
Note: I'll leave n = 5 to you
---------------------------------
e. Explain what the equation in part c tells you about Pascal�s triangle
The sum of the elements in the kth row is 2^(k-1)
====================================================
Cheers,
Stan H.
---------------------------------

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