Your question implies that 10 numbers have already been drawn from an unspecified source.
What you want to know is the probability of drawing the numbers again and getting 6 of the 10 in the same order as when they were first drawn. Is that right?
The first thing you need is the size of the source. Let N be the number of numbers there was to draw from.
For example if N=15 there are 3,003 ways of drawing 10 numbers out of 15, but if N=20 there are 184,756.
If N=10, all numbers would be drawn, but the number of permutations is 3,628,800 (10*9*8*...*2*1). The number of permutations of 6 out of 10 is 151,200 (10*9*8*7*6*5).
When making a comparison draw we need to know N to work out the probability of picking some of the same numbers in the same order.
Let's assume that N=10, and let's assume that we are going to measure the probability of drawing the first 6 numbers in the original draw order.
The chance of getting the first one to match the first in the original draw is 1/10, the second match is 1/9 because there are 9 left. The probability of matching first and second is 1/10*1/9=1/90. And so on.
So the probability of matching the 6 is 1/3628800. If we draw 10 numbers there are 210 ways (10*9*8*7*6*5/(6*5*4*3*2*1)) in which we can mix the 6 matched numbers and 4 others that may or may not match the correct order). We multiply the probability by 210=1/17,280
If N is bigger than 10 the probability decreases because not only do we have to get 6 order matches, but we have more numbers to choose from.
The probability of matching the first number drawn is 1/N, then the first two is 1/(N(N-1)) and so on.