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QUEST 2014: Homework 4
Pascal’s Triangle
Each problem is worth 4 points.
1. Draw the first five rows of Pascal’s Triangle (without looking at your
notes or the next page!)
2. Use Pascal’s Triangle to expand (x + y)5 .
3. Insert the values x = 1, y = 1 into the identity you obtained in
the previous problem. What identity do you obtain? (Your answer
should involve Pascal’s triangle.)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
,
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
4. Use Pascal’s Triangle to expand (x − 1)6 .
5. a) The numbers in the fourth diagonal of Pascal’s triangle, 1, 4, 10,
20,... are called what?
b) Why they are called this?
6. How many tennis balls would be required to make a tetrahedron with
6 balls along each edge? Use Pascal’s Triangle to answer this and
explain how you are using it.
7. Use the Hockey Stick Identity to find the sum 1 + 5 + 15 + 35 + 70,
and explain how you are using it.
8. What is the sum of the numbers in the 100-th row of Pascal’s Triangle? (The number at the top of Pascal’s Triangle does not count as a
row.)
9. a) What is the sum of the squares of the numbers in the fifth row
(1,5,10,10,5,1) of Pascal’s Triangle? (Just calculate this directly.)
b) Explain (in words) what number in Pascal’s Triangle this sum
equals.
10. a) Give the first four numbers in the 13-th row of Pascal’s Triangle.
(Use the formula’s for triangle and tetrahedral numbers.)
b) What common factor does every number in the 13-th row of Pascal’s Triangle have (except for the two 1’s).