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21 February 2008
Section 17
1.
Evaluate
3
2
±±
and
2
3
±±
by explicitly listing all possible multisets of the
appropriate size. Check that your answers agree with the formula in The
orem
17.8.
For
3
2
±±
the multisets are
h
1
;
1
i
,
h
2
;
2
i
,
h
3
;
3
i
,
h
1
;
2
i
,
h
1
;
3
i
,
h
2
;
3
i
There are six of them, and the formula in Theorem 17.8 gives
²²
3
2
³³
=
²
4
2
³
= 6
For
2
3
±±
the multisets are
h
1
;
1
;
1
i
,
h
1
;
1
;
2
i
,
h
1
;
2
;
2
i
,
h
2
;
2
;
2
i
There are four of them, and the formula in Theorem 17.8
²²
2
3
³³
=
²
4
3
³
= 4
2.
Give the starsandbars representation for all the sets you found in the
previous problem
.
h
1
;
1
i
h
2
;
2
i
h
3
;
3
i
h
1
;
2
i
h
1
;
3
i
h
2
;
3
i
h
1
;
1
;
1
i
h
1
;
1
;
2
i
h
1
;
2
;
2
i
h
2
;
2
;
2
i
4.
What multiset is encoded by the starsandbars notation
?
There are four stars, so we are choosing a multiset of size
4
. There are
three bars, so we are choosing from a set of size
4
, which we might as well
take to be
f
1
;
2
;
3
;
4
g
. So the indicated multiset is
h
1
;
4
;
4
;
4
i
. There is one
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 Spring '08
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