JSS 16 (2) (1960) 147-151
A NOTE ON SOME ELEMENTARY DERANGEMENT AND ALLIED PROBLEMS
by
M. T. L. BIZLEY
INTRODUCTION
T H E following method of solving a certain type of problem in
permutations and combinations, and in probability, is so simple
that it must have been used before. On the other hand, the writer
has never seen it expounded in the following form, and it does not
seem to be well known to students. Whilst, therefore, no originality
is claimed, this note may be found of interest to readers; and they
may care to compare the length of the solutions to the examples
below with those to be found in the text-books employing more
conventional methods.
THE INVERSION THEOREM
The method depends upon the fact that if, for all non-negative
integral values of s,
where Fi does not depend upon s if
then
The proof of this statement is elementary, but for the sake of completeness it is included as an Appendix to this note. The reader
will have no difficulty in extending the proof to show, more
generally, that if
then
148
M. T. L. BIZLEY
where
and
=
2
J =j1 +j2+ j
s1+
n
• • •
ILLUSTRATIVE EXAMPLES
(1) Derangements. In how many ways can s letters be put into
their s envelopes, one in each, with the restriction that no
letter goes into its own envelope? (Cf. Bizley, 1957, p. 79.)
Solution. Let F, be the required number. The s! ways of putting
the letters into the envelopes without the restriction may be
classified according to the number, i, of letters which do not go into
their own envelopes. That is,
since we can choose the (s—i) letters to go into their own envelopes
in (s.) ways and put the other i letters into wrong envelopes in Ft
ways, by definition. Hence from the Theorem, with U8 = s\,
(2) Distributions. To find the number of ways of distributing m
distinguishable objects among s distinguishable cells, with the
restriction that no cell is to be empty. (Cf. Bizley, 1957,
PP- 33. 83.)
Solution. Let F 8 ,m be the required number. Then the sm ways
of distributing the objects without the restriction can be classified
according to the number, i, of occupied cells. That is
with the convention
Noting that m is a mere parameter, we have at once from the
Theorem
ELEMENTARY DERANGEMENT AND ALLIED PROBLEMS 149
(3) Pairing. In how many ways can 2s people be arranged in pairs
(without regard to the order of the pairs or to the order of
people within a pair) with the restriction that nobody has the
same partner as in an earlier pairing arrangement? (Cf.
Bizley, 1957, p. 80.)
Solution. Let F8 be the required number. The (2s)!/*!(2!)*
ways of arrangement without the restriction can be classified
according to the number, i, of pairs which did not appear in the
earlier arrangement. As there are(s).1 ways of choosing these pairs
and Ft ways of arranging the 2i people into new pairs, we have
whence from the Theorem
and since
this may be written
The above examples are all simple well-known problems whose
solutions can be obtained by a variety of classical methods. Our
last example illustrates the application of our method to a less
elementary problem which is, nevertheless, well known (e.g.
Joseph & Bizley, i960).
(4) More general derangements. s1 letters are all written to the same
person and the same address, and s1 corresponding and
exactly similar envelopes are prepared. s2 letters are written
to another person and s2 envelopes (all addressed to him) are
prepared; and so on until finally sn letters and their sn similar
150
M. T. L. BIZLEY
envelopes are addressed to an nth person. If now the
letters are put into the
envelopes, in how many ways can
every letter go into a wrongly addressed envelope? (The first
example is, of course, the special case where each 's' is equal
to unity, and our 'n' is replaced by V.)
Solution. The (Hs)\ls1ls2\...snl ways of putting the letters into
the envelopes without restriction can be classified according to the
number jj of letters of the first set which do not go into a correct
envelope, the number i2 of letters of the second set which do not,
and so on. Since there are I } 11.21... I . n I ways of making the choice,
we must have, if F(s1,s2,•••,sn) is the required number,
and hence by the extension of the Theorem the required number is
APPENDIX
Proof of the Theorem
We have by hypothesis for all non-negative integral values of /,
ELEMENTARY DERANGEMENT AND ALLIED PROBLEMS 151
Hence
since
unless
when its value is unity. Alternatively, the result is very easy to
prove by finite difference methods.
REFERENCES
BIZLBY, M. T. L. (1957). Probability: An Intermediate Textbook. Cambridge
University Press.
JOSEPH, A. W. & BIZLEY, M. T. L. (i960). The two-pack matching problem.
J.R. Statist. Soc. (B) 22, no. 1, 114.