2008 iTest Problems/Problem 78
Contents
Problem
Feeling excited over her successful explorations into Pascal's Triangle, Wendy formulates a second problem to use during a future Jupiter Falls High School Math Meet:
How many of the first 2010 rows of Pascal's Triangle (Rows 0 through 2009) have exactly 256 odd entries?
What is the solution to Wendy's second problem?
Solution
Building off of Method 2 in problem 77, we see that each that the base 2 representation of the number reveals to us how many times we end up applying the
recursion relation 
.
For example, 
has seven nonzero terms in its base 2 expansion so that we end up
applying the recursion relation seven times giving a final term of 
. So we need to count the number of integers between 
which
have exactly 8 nonzero terms in their base 2 expansion.
.
counts the numbers up to 
which have exactly 8 nonzero terms
in their base 2 expansion. Of these 
candidates, the following 12 must be excluded since they exceed 
:
This gives an answer of 
.
Solution 2
By Lucas' Theorem, we require there to be 
ones in the binary representation of the row number. Continue as in solution 1.
See Also
| 2008 iTest (Problems) | ||
| Preceded by: Problem 77 |
Followed by: Problem 79 | |
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