Why Read This Book? - Index of

4.10 The Binomial Theorem 157
EXERCISE 4.9.31 The U.S. Senate consists of 100 senators, with 2 from each
state. In how many ways may a committee of 5 senators be formed so that no
2 senators are from the same state?
EXERCISE 4.9.32 You have 12 nieces and nephews, 3 of whom are kids of your
pyromaniac brother Torch. If you distribute 5 quarters among the 12 kids so that
no child receives more than one quarter, how many such distributions have at
least one of Torch’s kids receiving a quarter?
EXERCISE 4.9.33 You have 4 novels, 6 comic books, and 9 textbooks on your
bookshelf. In how many different ways can you arrange them if you want the
books of each type to be grouped together?
4.10 The Binomial Theorem
The appearance of combinations in a variety of settings reveals some nice ties
between seemingly unrelated mathematical ideas. First, there is a nice way to
visualize Exercise 4.9.2 in Pascal’s triangle (Fig. 4.8). To construct Pascal’s triangle,
we first extend the definition of C(n, k) to allow for kn. In both of
these cases we define C(n, k) = 0.
EXERCISE 4.10.1 Verify that Eq. (4.26) holds for k ≤ 0 and k ≥ n + 1.
To construct Pascal’s triangle, we begin with a row that corresponds to combinations
of the form C(0,k), which we will therefore call row zero. Because
C(0, 0) = 1 and C(0,k)= 0 for all nonzero values of k, we construct row 0 to have
a single 1 entry, with 0s elsewhere.
The next row corresponds to n = 1, and so on, and each entry in a row is
determined by adding the two entries diagonally above it in the previous row.
n 5 0
n 5 1
n 5 2
n 5 3
n 5 4
n 5 5
n 5 6
Figure 4.8 Pascal’s triangle.
k 5 n k 5 0
0 0 0 0
0 0 1 0 0
0 1 1 0
0 1 2 1 0
0 1 3 3 1 0
0 1 4 6 4 1 0
0 1 5 10 10 5 1 0
0 1 6 15 20 15 6 1 0