Pascal's Triangle

[Treeclimber]

Mary is choosing 3 of the 9 contestestants. In how many ways can this be done?
Is the answer 9*8*7? And to find it on Pascal's triangle do I go 10 spaces down to the 9th row and 4 spaces across to the 3th element?

 

[Deleted member 4993]

Mary is choosing 3 of the 9 contestestants. In how many ways can this be done?
Is the answer 9*8*7? And to find it on Pascal's triangle do I go 10 spaces down to the 9th row and 4 spaces across to the 3th element?

Since order does not matter here - the proper way would be to use combination:

\(\displaystyle _nC_r \ \ = \ \ \frac{9!}{3!\cdot (9-3)!} \ \\)

 

[soroban]

Hello, Treeclimber!

Mary is choosing 3 of the 9 contestestants.
In how many ways can this be done?

Is the answer 9*8*7 ? . . . . no

And to find it on Pascal's triangle do I go 10 spaces down to the 9th row and 4 spaces across to the 3th element?
. . Um ... no

. . \(\displaystyle \begin{array}{cc} 1 \\ 1\quad 1 \\ 1 \quad 2 \quad 1 \\ 1 \quad 3 \quad 3 \quad 1 \\ 1 \quad 4 \quad 6 \quad 4 \quad 1 \\ \vdots \\ 1 \quad 8 \quad 28 \quad 56 \quad 70 \quad 56 \quad 28 \quad 8 \quad 1 \\ 1 \quad 9 \quad 36 \quad 84 \quad 126 \quad 126 \quad 84 \quad 36 \quad 9 \quad 1 \end{array}\)
. . . . . . . . . . \(\displaystyle \uparrow\)

You go 9 spaces down to the 10th row and 3 spaces across to the 4th element.

 

[carroncampbell]

is there a way of predicting the line of the triangle,without having to work out each number by adding the two numbers above it?how?
this go up to ten row

 

[carroncampbell]

calculate the totals of the numbers in each horizontal row,is there a pattern?can you predict the next total?

 

[tkhunny]

Predict? What a strange word. How about "calculate"?

There are many patterns in the triangle. Try the "Hockey Stick" rule. I explain one way:

1) Start anywhere on the right slope.
2) Move parallel to the left slope and add up as you go.
3) Whenever you stop, the total is on the next row and to the right. :wink:

Figure out how to determine the number of odd numbers in each row! :?

 

[TchrWill]

is there a way of predicting the line of the triangle,without having to work out each number by adding the two numbers above it?how?
this go up to ten row

How can you determine the numbers in any row without knowing the numbers in the previous row?

Let the row number start from 0 at the top. Call that number n. Let the number of the number in the row be r, starting from r=1 at the left.

The value of the number that goes in the rth place of that row is

n!/[r! (n-r)!]]

Consider, for example, the fifth row of the triangle that has elements 1 4 6 4 1

n=4 in this case, since we started with n=0 at the top. For the second element, r=2
The value of that number is
4!/(2! 2!) = 3*4/1*2 = 6