1.4.1 Sampling without Replacement Flashcards
(22 cards)
What does ‘sampling without replacement’ mean?
Once you pick an item, it’s gone — no take-backs. You don’t put it back in the set.
Why is sampling without replacement important in counting?
Because it affects how many choices you have at each step — each pick shrinks the pool.
What is the definition of a factorial?
$n! = n \cdot (n-1) \cdot (n-2) \cdots 1$, with $0! = 1$
What does $n!$ represent?
The number of ways to arrange $n$ distinct elements in order.
If you have 3 digits (1, 2, 3), how many three-digit numbers can you make using all of them once?
3! = 6
What is a permutation?
An arrangement of items where order matters.
What is the formula for the number of r-permutations of n items?
$$ P(n, r) = \frac{n!}{(n - r)!} $$
When do you use permutations?
When the order of selection matters, and there’s no replacement.
How many 2-digit numbers can be made from digits {2,3,4,5} with no repeats?
P(4, 2) = 4 * 3 = 12
What’s the main difference between combinations and permutations?
In combinations, order does NOT matter.
What is the formula for the number of r-combinations of n items?
$$ \binom{n}{r} = \frac{n!}{r!(n-r)!} $$
Why divide by $r!$ in combinations?
To account for overcounting all the rearrangements of the same group.
How many 2-sock pairs can you make from 5 socks?
$\binom{5}{2} = 10$
If order doesn’t matter and there’s no replacement, which technique do you use?
Combination.
What is a partition in probability?
Dividing $n$ items into disjoint groups whose union is the whole set.
What is the formula for partitioning $n$ items into $k$ groups of sizes $n_1, n_2, …, n_k$?
$$ \frac{n!}{n_1! \cdot n_2! \cdots n_k!} $$
Why divide by all the $n_i!$’s in a partition?
To remove overcounting due to identical groupings.
What if some elements are repeated (e.g., 3 A’s and 2 B’s)? How do you count distinct permutations?
Use partitioning: $$ \frac{n!}{r_1! \cdot r_2! \cdots} $$
When should you use factorials?
When arranging all $n$ distinct items in order.
When should you use permutations?
When arranging some $r \leq n$ items, where order matters.
When should you use combinations?
When selecting items without regard to order.
When should you use partitions?
When breaking items into multiple unordered groups, or when items are not all distinct.
