SAMPLING WITH REPLACEMENT

[abhisheksharma12593Participant]

Note: Sampling with Replacement. The counting method described in Example
1.8.4 is a type of sampling with replacement that is different from the type
described in Example 1.7.10. In Example 1.7.10, we sampled with replacement, but
we distinguished between samples having the same balls in different orders. This
could be called ordered sampling with replacement. In Example 1.8.4, samples containing
the same genes in different orders were considered the same outcome. This
could be called unordered sampling with replacement. The general formula for the
number of unordered samples of size k with replacement from n elements is
n+k−1
k

,
and can be derived in Exercise 19. It is possible to have k larger than n when sampling
with replacement.
Example
1.8.5
Selecting Baked Goods. You go to a bakery to select some baked goods for a dinner
party. You need to choose a total of 12 items. The baker has seven different types
of items from which to choose, with lots of each type available. How many different
boxfuls of 12 items are possible for you to choose? Here we will not distinguish the
same collection of 12 items arranged in different orders in the box. This is an example
of unordered sampling with replacement because we can (indeed we must) choose
the same type of item more than once, but we are not distinguishing the same items
in different orders. There are
7+12−1
12

= 18,564 different boxfuls.

CAN ANYONE TELL WHY WE TAKE n+k-1(C)k

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