Homework Help???

Question

My math homework says to find how the fourth row of the Pascal's Triangle has to do with a real life situation. The fourth row is 1,3,3,1. Help!

Answer 1

Note, the rows of Pascal's triangle are usually counted starting at zero.

Row 0 = 1
Row 1 = 1 1
Row 2 = 1 2 1
Row 3 = 1 3 3 1
Row 4 = 1 4 6 4 1

One way is in figuring the ways to toss coins and get a certain number of heads.

For example, toss 1 coin.
There is 1 way to get 0 heads.
There is 1 way to get 1 head.

This is the same as row 1 1 of Pascal's triangle.

Now try two coins.
T T = 0
T H = 1
H T = 1
H H = 2

So there is 1 way to get 0 heads.
There are 2 ways to get 1 head.
There is 1 way to get 2 heads.

This is the same as row 1 2 1 of Pascal's triangle.

Now with three coins you have:
T T T = 0
T T H = 1
T H T = 1
T H H = 2
H T T = 1
H T H = 2
H H T = 2
H H H = 3

So there is 1 way to get 0 heads.
There are 3 ways to get 1 head.
There is 3 ways to get 2 heads.
There is 1 way to get 3 heads.

This is the same as row 1 3 3 1 of Pascal's triangle.

Continuing with four coins, you'll find you have:
1 way to get 0 heads
4 ways to get 1 head
6 ways to get 2 heads
4 ways to get 3 heads
1 way to get 4 heads

This is the same as row 1 4 6 4 1 of Pascal's triangle.

Another match to real life is in connecting points. Draw 4 points. If you connect them in all possible ways, you get 6 lines. If you count the number of triangles formed by these lines you get 4. And if you count the number of quadrilaterals (a square) you get 1. This matches with the pattern 1 4 6 4 1.

Let's also look at (x + 1)^4...

This becomes (x + 1)(x + 1)(x + 1)(x + 1)
If you expand it out you get:

x^4 + 4 x^3 + 6 x^2 + 4 x + 1

Look at the coefficients and you have 1 4 6 4 1 again!

Or imagine you had four letters (A, B, C, D). How many ways are there to create subsets of these letters?

{ } = 0 elements = 1 way
{A}, {B}, {C}, {D} = 1 elements = 4 ways
{AB}, {BC}, {CD}, {AC}, {AD}, {BD} = 2 elements = 6 ways
{ABC}, {ABD}, {ACD}, {BCD} = 3 elements = 4 ways
{ABCD} = 4 elements = 1 way

This is again the 1 4 6 4 1 pattern...

Is this enough?

Answer 2

The first row is really called row "0" so the 4th row is:
1 4 6 4 1.

One of the properties of pascal's triangle is that 11 to the power of the row number = the numbers in the row:

11^0 = 1
11^1 = 11
11^2 = 121
11^3 = 1331
11^4 = 14641
Hope this helps.

Answer 3

-1 - 3 - 3 - 1-
1- 4 - 6 - 4 - 1 ( 4th row )
1- 5 - 10 - 1-5 - 1 fifth row

4 = 1+3. 6 = 3+3