46 3. Binomial Coefficients and Pascal’s Triangle
(a)n=k,n 1 =n 2 =···=nk=1;
(b) n 1 =n 2 =···=nk− 1 =1,nk=n−k+1;
(c) k=2;
(d) k=3,n=6,n 1 =n 2 =n 3 =2.FIGURE 3.1. Placing 8 nonattacking rooks on a chessboard.3.2.3 (a) How many ways can you placenrooks on a chessboard so that no
two attack each other (Figure 3.1)? We assume that the rooks are identical,
so interchanging two rooks does not count as a separate placement.
(b) How many ways can you do this if you have 4 wooden and 4 marble rooks?
(c) How many ways can you do this if all the 8 rooks are different?3.3 Anagrams............................
Have you played with anagrams? One selects a word (say, COMBINA-
TORICS) and tries to compose from its letters meaningful or, even better,
funny words or expressions.
How many anagrams can you build from a given word? If you try to an-
swer this question by playing around with the letters, you will realize that
the question is badly posed; it is difficult to draw the line between meaning-
ful and nonmeaningful anagrams. For example, it could easily happen that
A CROC BIT SIMON. And it may be true that Napoleon always wanted
a TOMB IN CORSICA. It is questionable, but certainly grammatically
correct, to assert that COB IS ROMANTIC. Some universities may have
a course on MAC IN ROBOTICS.