Question

question 1 (1 point) four puppies are selected at random for a commercial from a dog breeder that has 26 puppies all of different weights in grams. what is the probability that they will be in order of weight? a) $\frac{4!}{_{26}p_{4}}$ b) $\frac{1}{26^{4}}$ c) $\frac{4}{26!}$ d) $\frac{1}{_{26}p_{4}}$

Explanation:

Step1: Calculate total arrangements

The number of permutations of selecting 4 puppies out of 26 is given by the permutation formula $_{n}P_{r}=\frac{n!}{(n - r)!}$, so $_{26}P_{4}=\frac{26!}{(26 - 4)!}=\frac{26!}{22!}=26\times25\times24\times23$. This is the total number of ways to arrange 4 - puppy selections.

Step2: Determine favorable arrangements

The number of ways to arrange the 4 selected puppies in order of weight (either ascending or descending) is 2 (either from lightest - to - heaviest or heaviest - to - lightest). But if we consider only one - direction (say ascending), there is only 1 way to arrange them in order of weight among all the possible arrangements of the 4 selected puppies.

Step3: Calculate probability

The probability $P$ of an event is the number of favorable outcomes divided by the number of total outcomes. So the probability that the 4 selected puppies are in order of weight is $\frac{1}{_{26}P_{4}}$.

Answer:

D. $\frac{1}{_{26}P_{4}}$