Identifying the pattern - 1

(10/1),(45/10=9/2),(120/45=8/3),(210/120=7/4),(x/210=6/5) therefore x=6*210/5=252

$\binom{10}{0}, \binom{10}{1}, \binom{10}{2},......, \binom{10}{9}, \binom{10}{10}$

So the desired number is $\binom{10}{5}=\boxed{252}$

Rule: To get any number from the previous one: Multiply current number by (11 - current position) then divide by (current position) : Here's how it works: To get 2nd number: 1 X (11 - 1) / 1 = 10 To get 3rd number: 10 X (11 - 2) / 2 = 45 To get 4th number: 45 X (11 - 3) / 3 = 120 To get 5th number: 120 X (11 - 4) / 4 = 210 To get 6th number (the missing one) : 210 X (11 - 5) / 5 = 252 We can also continue and we'll get the same numbers in decreasing order:
To get 7th number: 252 X (11 - 6) / 6 = 210 To get 8th number: 210 X (11 - 7) / 7 = 120 To get 9th number: 120 X (11 - 8) / 8 = 45 To get 10th number: 45 X (11 - 9) / 9 = 10 And finally, To get 11th number: 10 X (11 - 10) / 10 = 1

Here's how i did it

$1-10+45-120+210+210-120+45-10+1$

$\boxed{252}$

210*6/5 = 252. Binomial expansion theorem aka Pascal's Triangle.

Note that these are binomial coefficients. In fact, they are of the form $\binom{10}{n}$ . The answer is $\binom{10}{5}=\boxed{252}$ .

1x10/1=10_ 10x9/2=45_ 45x8/3=120_ 120x7/4=210_ 210x6/5=252

This is just the 11th row in Pascal's Triangle