Help with Pascal’s Triangle: Paths and Binary Strings
a. The additional paths between the starting point (number 1 at the top) and
the ending point (leftmost 3) as shown in the given diagram are
1. Going down left, then down right and finally down left. The corresponding binary string is 010.
.
2. Going down right, then down left and again down left. The
corresponding binary string is 100.
b. Consider the leftmost 4 in the 4th row. The paths are
.
Consider the middle 6 in the 4th row. The paths are
.
Consider the rightmost 4 in the 4th row. The paths are
.
Consider the leftmost 5 in the 5th row. The paths are
.
Consider the rightmost 5 in the 5th row. The paths are
.
Similarities and differences:
The number of digits in a binary string of n-th row is n. If we
ignore the border line 1s of the triangle and begin counting from
the left, then number of 1s in the binary string in the m-th number
of a row is m.
Also, the number of paths for a particular entry
is k.
c. As already stated in (b) , the length of a binary string of a
number in n-th row is n. The number of 1s in the binary string in
the m-th number of a row is m.
d. Consider two entries and in a particular row. Adding them, one will get the entry in the next row.
Hence number of paths leading to the entry in a particular row is the sum of the numbers of paths leading to the two entries and in the previous row.
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