There are four odd numbers in the 6th row of Pascal's triangle (1, 15, 15, 1), and 4 = 22 is a power of 2. For the 0th through 12th rows of Pascal's triangle, show that the number of odd numbers in each row is a power of 2.
In Fig. 1(a), each odd number in the first eight rows of Pascal's triangle has been replaced by a dot and each even number has been replaced by a space. In Fig. 1(b), the pattern for the first four rows is shown in blue. Notice that this pattern appears twice in the next four rows, with the two appearances separated by an inverted triangle. Fig. 1(c) shows the locations of the odd numbers in the first 16 rows of Pascal's triangle, and Fig. 1(d) shows that the pattern for the first eight rows appears twice in the next eight rows, separated by an inverted triangle. Figure 2 demonstrates that the first 32 rows of Pascal's triangle have the same property.
Figure 1

Figure 2

In the following triangular table, known as Pascal's triangle, the entries in the nth row are the binomial coefficients

Observe that each number (other than the ones) is the sum of the two numbers directly above it. For example, in the 5th row, the number 5 is the sum of the numbers 1 and 4 from the 4th row, and the number 10 is the sum of the numbers 4 and 6 from the 4th row. This fact is known as Pascal's formula. Namely, the formula says that

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(c) (a) (b) (d) (b)