From a group of n people, suppose that we want to choose a committee of k , k ≤ n , one of whom is to be designated as chairperson. a. By focusing first on the choice of the committee and then on the choice of the chair, argue that there are ( n k ) k possible choices. b. By focusing first on the choice of the nonchair committee members and then on the choice of the chair, argue that there are ( n k − 1 ) ( n − k + 1 ) possible choices. c. By focusing first on the choice of the chair and then on the choice of the other committee members, argue that there are ( n − 1 k − 1 ) possible choices. d. Conclude from parts (a), (b), and (C) that k ( n k ) = ( n − k + 1 ) ( n k − 1 ) = n ( n − 1 k − 1 ) e. Use the factorial definition of ( m r ) to verify the identity in part (d).
From a group of n people, suppose that we want to choose a committee of k , k ≤ n , one of whom is to be designated as chairperson. a. By focusing first on the choice of the committee and then on the choice of the chair, argue that there are ( n k ) k possible choices. b. By focusing first on the choice of the nonchair committee members and then on the choice of the chair, argue that there are ( n k − 1 ) ( n − k + 1 ) possible choices. c. By focusing first on the choice of the chair and then on the choice of the other committee members, argue that there are ( n − 1 k − 1 ) possible choices. d. Conclude from parts (a), (b), and (C) that k ( n k ) = ( n − k + 1 ) ( n k − 1 ) = n ( n − 1 k − 1 ) e. Use the factorial definition of ( m r ) to verify the identity in part (d).
From a group of n people, suppose that we want to choose a committee of
k
,
k
≤
n
, one of whom is to be designated as chairperson.
a. By focusing first on the choice of the committee and then on the choice of the chair, argue that there are
(
n
k
)
k
possible choices.
b. By focusing first on the choice of the nonchair committee members and then on the choice of the chair, argue that there are
(
n
k
−
1
)
(
n
−
k
+
1
)
possible choices.
c. By focusing first on the choice of the chair and then on the choice of the other committee members, argue that there are
(
n
−
1
k
−
1
)
possible choices.
d. Conclude from parts (a), (b), and (C) that
k
(
n
k
)
=
(
n
−
k
+
1
)
(
n
k
−
1
)
=
n
(
n
−
1
k
−
1
)
e. Use the factorial definition of
(
m
r
)
to verify the identity in part (d).
13) Consider the checkerboard arrangement shown below. Assume that the red checker can move diagonally
upward, one square at a time, on the white squares. It may not enter a square if occupied by another checker, but
may jump over it. How many routes are there for the red checker to the top of the board?
12) The prime factors of 1365 are 3, 5, 7 and 13. Determine the total number of divisors of 1365.
11) What is the sum of numbers in row #8 of Pascal's Triangle?
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