Question

answer the following question using the appropriate counting technique, which may be either arrangements with repetition, permutations, or combinations. be sure to explain why this counting technique applies to the problem.

how many possible birth orders with respect to sex are possible in a family of rabbits with a litter of six bunnies? (for example, mmmfff and fmfmfm are different orders.)

what counting technique should be used to make this calculation?

a. permutations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement matters.
b. arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
c. combinations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement does not matter.
d. arrangements with repetitions because there are r selections from a group of n choices and choices can be repeated.

there are possible birth orders for a family of rabbits with a litter of six bunnies.
(type a whole number.)

Explanation:

Identify the counting scenario

We analyze the birth order of six bunnies with respect to sex (Male or Female).
Each of the six positions in the birth order has \(2\) possible choices: Male (M) or Female (F).
Since the sex of one bunny does not restrict the sex of another, choices can be repeated.
The order of the arrangement matters (e.g., MMMFFF is different from FMFMFM).

Select the counting technique

Using the Arrangements with Repetitions knowledge point:
We make \(r = 6\) selections from a group of \(n = 2\) choices (Male, Female), where choices can be repeated.
Thus, the correct option is D.

Calculate the total possibilities

Using the Fundamental Counting Principle knowledge point:
The number of possible birth orders is:

$$ 2^6 = 64 $$

Answer:

Question 1

• (A) Permutations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement matters.
• (B) Arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
• (C) Combinations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement does not matter.
• (D) Arrangements with repetitions because there are r selections from a group of n choices and choices can be repeated. (Correct answer)

Question 2

There are <blank>64</blank> possible birth orders for a family of rabbits with a litter of six bunnies.