Question

Use Pascal's triangle to find the number of ways to choose 5 pens from 7 pens

Answer 1

do you have to write pascals triangle down to level 7? if not just compute
\[\dbinom{7}{5}=\frac{7\times 6}{2}=21\]

Answer 2

I am pretty sure im suppose to use Pascal's triangle written out and not the formula for this question bu i dont know what in Pascal's triangle would give me the soultion

Answer 3

elsewise you have to write out the whole triangle down to 7 levels
1
11
121
1331
14641
15 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1

and the 21 is both the third and 6th entry

Answer 4

How am i suppose to know that 21 is the answear doing it that way?

Answer 5

the 7th row of pascals triangle is
1 7 21 35 35 21 7 1
the first entry is 1, the number of ways you can choose no things out of 7
the second entry is 7, the number of ways you can choose 1 out of 7
21 is the number of ways to choose 2 out of 7
35 number of ways to choose 3
also 4
etc.

Answer 6

okay thank you that makes sense now

Answer 7

notice also that the number of ways to choose 5 out of seven is the same as the number of ways to choose 2. that is obvious because choosing 5 to include is the same as choosing 2 to exclude.

Answer 8

so the row with just the one is row zero?

Answer 9

Yeah, it helps to think of all the 1s as zero terms. Then for picking 5 objects out of 7, you're looking at entry 5 in row 7.

Answer 10

yes the top is the 'zero' level it is what you get if you expand
\[(a+b)^0\]
the next level gives
\[(a+b)^1=a+b\]
the next gives
\[(a+b)^2=a^2+2ab+b^2\]
and so on. so the '7th' level is the one that starts
1 7 ...