2020 AMC 10B Problems/Problem 25
Revision as of 10:50, 23 December 2021 by Isabelchen (talk | contribs)
Solution 7(Integer Partition)
Note that .
depends on dividing
into different terms, which is the integer partition of
.
Divide into
term:
There is only one way.
Divide into
terms:
Case:
is alone
has
different arrangements.
Case:
is with
![]()
. For
and
,
can be with any term from the
tuples, and the arrangement of the
terms is
.
![]()
Divide into
terms:
Case:
is alone
. For
and
, there are
arrangements each.
![]()
Case:
is with
![]()
. For
and
,
can be with any term from the
tuples. If
is with
for the first tuple, or
for the second tuple, the number of arrangements will be
for each. If
is with
for the first tuple, or
for the second tuple, the number or arrangements will be
for each.
![]()
Divide into
terms:
Case:
is alone
. For
and
, there are
arrangements each.
![]()
Case:
is with
![]()
. For
,
can be with any term from the tuple. If
is with
, the number of arrangements will be
. If
is with
, the number or arrangements will be
.
![]()
Divide into
terms:
When dividing into
parts there are
cases.
Case:
is alone
. For
, there are
arrangements.
![]()
Case:
is with
![]()
. For
,
can only be with
. The number or arrangements will be
![]()
Divide into
terms:
,
~isabelchen