Sequences 4: Pascal's Triangle

http://www.youtube.com/watch?v=bMB8qDYa8N0

 

 

What do you know about Pascal?

What do you know about Pascal’ s triangle?

 

 

Listen to and watch the video, answer questions and fill in the missing items.

 

 

a)      What is the aim of the lecture? …………………………………………………

b)      We need to recognize the relations between …………………………………..

c)      The first and the last terms are both …………………………………………..

d)      Whatever exponent I start with on the variable it ……………from that exponent down to an exponent of ………..The other exponent ……………from zero up to x to the nth power.

e)      The first and last …………….. are1.

f)        The so called binomial expansions keep you from ………………………………..

g)      You add two numbers above to get ……………………………………………….

h)      Pascal was a ………………………………………………………………………

i)        If you know Pascal’ s triangle, you can …………………………………………..

j)        If you add up …………………… of  Pascal’s triangle, you get Fibonacci sequence.

k)      What does the speaker like about maths? …………………………………………….   

 

 

 

 

 

 

 

 

 

 

 

 

Sequences 4: Pascal's Triangle

http://www.youtube.com/watch?v=bMB8qDYa8N0

 

 

What do you know about Pascal?

What do you know about Pascal’ s triangle?

 

 

Listen to and watch the video, answer questions and fill in the missing items.

 

 

l)        What is the aim of the lecture? …………………………………………………

m)    We need to recognize the relations between ………terms and exponents.

n)      The first and the last terms are both ……………to the fifth power…..

o)      Whatever exponent I start with on the variable it ……decreases from that exponent down to an exponent of …zero……..The other exponent ……increases…from zero up to x to the nth power.

p)      The first and last ……coefficients.. are1.

q)      The so called binomial expansions keep you from ………having to do multiplication..

r)       You add two numbers above to get …………the number below…….

s)       Pascal was a ……………French philosopher and mathematician……

t)        If you know Pascal’ s triangle, you can ………read numbers in Japanese…..

u)      If you add up ……diagonals… of  Pascal’s triangle, you get Fibonacci sequence.

v)      What does the speaker like about maths? …………it is related to st else…….    

 

 

 

 

Pascal´ s  Triangle

 

Adapted from Nucleus, Maths, English for Science and Technology

 

Read the text and fill in the missing words. Try to guess.

 

A single dice is thrown. There are two possible outcomes – odd or ……... The chances of throwing an odd number are 1 in 2, and so are the ……… of throwing an even number. Supposing now that two ……. are thrown. There are now three possible outcomes: both odd, both even, or one odd and one even. But the last result, one odd and one even, can occur in two different ways, either the …….. dice odd and the second dice even, or the first dice even and the second dice odd. So the chances of throwing ……. odd are 1 in 4 of throwing both even 1 in 4, and of throwing one odd and one even 2 in 4. Throwing three dice produces ……. different possibilities: EEE, EEO, EOO, EOE, OEE, ……, OOE and OOO. Thus the probabilities are as follows: all evens, 1 in 8, all odds 1 in 8, two odds and one even, …. in 8, two evens and one ……, 3 in 8.

 

    These results can be ………. by using a device called Pascal´ s Triangle, the first three rows of which are shows in Fig. 10.1. This triangle can easily be formed. The first and last figure in each row is ……. Every other figure is the ……. of the two figures above it. Thus, in the second row, 2 is the sum of 1 and 1. In the third row, both threes are formed by ……… 2 and 1. The total of all figures in each row gives the total number of possibilities for that row. Thus the third row has 1+…+3+1 possibilities.  

 

 

 

 

 

A single dice is thrown. There are two possible outcomes – odd or even. The chances of throwing an odd number are 1 in 2, and so are the chances of throwing an even number. Supposing now that two dice are thrown. There are now three possible outcomes: both odd, both even, or one odd and one even. But the last result, one odd and one even, can occur in two different ways, either the first dice odd and the second dice even, or the first dice even and the second dice odd. So the chances of throwing both odd are 1 in 4 of throwing both even 1 in 4, and of throwing one odd and one even 2 in 4. throwing three dice produces eight different possibilities: EEE, EEO, EOO, EOE, OEE, OEO, OOE and OOO. Thus the probabilities are as follows: all evens, 1 in 8, all odds 1 in 8, two odds and one even, 3 in 8, two evens and one odd, 3 in 8.

 

    These results can be calculated by using a device called Pascal´ s Triangle, the first three rows of which are shows in Fig. 10.1. This triangle can easily be formed. The first and last figure in each row is one. Every other figure is the sum of the two figures above it. Thus, in the second row, 2 is the sum of 1 and 1. In the third row, both threes are formed by adding 2 and 1. the total of all figures in each row gives the total number of possibilities for that row. Thus the third row has 1+3+3+1 possibilities.