Question

the \triangle\ that pascal studied and published in his treatise was actually more like a truncated corner of tartaglias rectangle. each number in tartaglias rectangle can be calculated in various ways. consider the number n to be located anywhere in the array. by checking several locations in the given array, determine how n is related to the sum of all entries in the shaded cells. click the icon to view the truncated corner of tartaglias rectangle. complete the following table of the value of n and the sum of the values of the shaded cells for various locations of n in the truncated corner of tartaglias rectangle. location of n value of n sum of values of shaded cells row 8, column 2 8 8 row 6, column 3 21 21 row 4, column 4

Explanation:

Step1: Recall Pascal - like triangle properties

In a Pascal - like triangle (Tartaglia's rectangle in this case), the numbers follow a certain combinatorial pattern. The number in the $n$th row and $k$th column of the Pascal's triangle is given by the binomial coefficient $\binom{n - 1}{k - 1}$.

Step2: Calculate for Row 4, Column 4

For the $n$th row and $k$th column of the Tartaglia's rectangle, the number $N=\binom{n - 1}{k - 1}$. Here, $n = 4$ and $k = 4$. Using the binomial - coefficient formula $\binom{a}{b}=\frac{a!}{b!(a - b)!}$, when $a=n - 1=3$ and $b = k - 1=3$, we have $\binom{3}{3}=\frac{3!}{3!(3 - 3)!}=\frac{3!}{3!0!}=1$.
The sum of the values of the shaded cells is equal to the value of $N$. So the sum of the values of the shaded cells is also 1.

Answer:

Row 4, Column 4: Value of N is 1, Sum of values of shaded cells is 1