Permutations and Combinations

3.6 Lots of Practice

All solutions appear in the COMPANION GUIDE to this Permutations and Combinations course.  

 

Exercise 35: Compute the powers\(101^{0}\), \(101^{1}\), \(101^{2}\), \(101^{3}\), \(101^{4}\) and \(101^{5}\). What do you notice? Explain what you notice.

 

Exercise 36: Without a calculator compute \(201^{6}\).

 

Exercise 37: a) Without doing a lick of arithmetic explain why

\(64 – 6 \cdot 32 + 15 \cdot 16 – 20 \cdot 8 + 15 \cdot 4 – 6 \cdot 2 + 1\)

must equal \(1\).

 

b) Say something interesting that can be deduced from examining \((3 -1)^{6}\).

 

 

Exercise 38:

Here’s Pascal’s triangle:

P26001

The stocking property can be interpreted as follows:

Choose any entry “1” on the side of the triangle and follow the diagonal from that entry into the interior of the triangle. Turn downwards ninety degrees to form the toe of the stocking. Then the number in the toe equals the sum of the numbers in the leg of the stocking.

 

 Convince yourself – again – that this property is valid.

 

a)    The entries on the right-most diagonal of Pascal’s triangle are:  1 1 1 1 1 1…

P26002

Explain why the stocking property tells us that the entries of the next diagonal of Pascal’s triangle must be: 1 2 3 4 5 ….

 

b)    Knowing that the second diagonal of Pascal’s triangle has entries 1 2 3 4 5 6 …

P26003

explain, using the stocking property, why the entries of next diagonal of Pascal’s triangle must be the triangular numbers: 1 3 6 10 15 21 ….

 

c)     Use the stocking property to quickly determine the sum of the first six triangle numbers.

 

d)    The sum of the first six triangle numbers is called the sixth tetrahedral number. What is a tetrahedron? Why is this name appropriate?

 

e)    Use Pascal’s triangle to find the sum of the first six tetrahedral numbers.

 

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