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Another way to count the total number of all committees with a chair, is to consider Frst
selecting the chairperson from which we have
n
choices and then considering all possible
subsets of size
n

1(wh
i
chi
s2
n

1
)f
romwh
i
chtocon
s
t
ru
c
tth
er
ema
in
ingcomm
i
t
t
e
e
members. The product then gives
n
2
n

1
.
Part (b):
Consider again
n
people where now we want to count the total number of com
mittees of size
k
with a chairperson and a secretary. We can select all subsets of size
k
in
±
n
k
²
ways. Given a subset of size
k
,thereare
k
choices for the chairperson and
k
choices
for the secretary giving
k
2
±
n
k
²
committees of size
k
with a chair and a secretary. The
total number of these is then given by summing this result or
n
³
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 Winter '08
 G.JOGESHBABU
 Probability

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