## whatisthequestion 4 years ago Use Pascal's triangle to find the number of ways to choose 5 pens from 7 pens

1. satellite73

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

2. whatisthequestion

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

3. satellite73

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

4. whatisthequestion

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

5. satellite73

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.

6. whatisthequestion

okay thank you that makes sense now

7. satellite73

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.

8. whatisthequestion

so the row with just the one is row zero?

9. 3pwood

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.

10. satellite73

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 ...