Question

over the years, many interesting patterns have been discovered in pascals triangle. explore a few of them in the problem below. refer to pascals triangle and complete (a) through (c) below. click the icon to view the first 10 rows of pascals triangle. (a) choose a row whose row number is prime. except for the 1s in this row, what is true of all the other entries? (b) choose a second prime row number and see if the same pattern holds. (c) use the usual method to construct row 11 in pascals triangle. verify that the same pattern holds. in the answer boxes to complete your choice. a. row 11 has 11 values. row 11 b. row 11 has 10 values. row 11 c. row 11 has 12 values. row 11 1 11 55 165 330 462 462 330 165 55 11 1 d. row 11 cannot be constructed. the same pattern for row 11.

Explanation:

Step1: Recall property of Pascal's triangle rows

The number of values in the \(n\) -th row of Pascal's triangle is \(n + 1\). For row \(n=11\), the number of values is \(11 + 1=12\).

Step2: Construct row 11

We know that to construct a row of Pascal's triangle, we use the formula \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\), where \(n\) is the row - number and \(k\) ranges from \(0\) to \(n\). For \(n = 11\), we have:
\(\binom{11}{0}=1\), \(\binom{11}{1}=\frac{11!}{1!(11 - 1)!}=\frac{11!}{1!×10!}=11\), \(\binom{11}{2}=\frac{11!}{2!(11 - 2)!}=\frac{11\times10}{2\times1}=55\), \(\binom{11}{3}=\frac{11!}{3!(11 - 3)!}=\frac{11\times10\times9}{3\times2\times1}=165\), \(\binom{11}{4}=\frac{11!}{4!(11 - 4)!}=330\), \(\binom{11}{5}=\frac{11!}{5!(11 - 5)!}=462\), \(\binom{11}{6}=462\) (due to symmetry \(\binom{n}{k}=\binom{n}{n - k}\)), \(\binom{11}{7}=330\), \(\binom{11}{8}=165\), \(\binom{11}{9}=55\), \(\binom{11}{10}=11\), \(\binom{11}{11}=1\).
So row 11 is \(1,11,55,165,330,462,462,330,165,55,11,1\).

Step3: Analyze the pattern for prime - numbered rows

In a prime - numbered row \(n\) of Pascal's triangle (except for the \(1\)s at the ends), all the other entries are divisible by \(n\). For \(n = 11\), \(11\) divides \(11,55,165,330,462\) (since \(55\div11 = 5\), \(165\div11=15\), \(330\div11 = 30\), \(462\div11 = 42\)).

Answer:

C. Row 11 has 12 values.
Row 11: 1 11 55 165 330 462 462 330 165 55 11 1
The same pattern holds.