Question

astronomers have discovered a large asteroid which will strike earth in 8 years, obliterating civilization and throwing humanity into a dark age of brutality and cannibalism. nasa must select an emergency team of 10 to land on the asteroid and bury nukes inside it. knowing the importance of a good photo - op, nasa prioritizes equal gender representation as the selection criteria of the landing team, which will consist of 5 women and 5 men. what of the following expressions represents the number of different landing teams that could be selected from these 80 candidates? (30c5)(50c5) (30p5)(50p5) (50c5)(50c5) (5c30)(5c50) (5p30)(5p50) question help: video check answer

Explanation:

Step1: Recall combination formula

The number of ways to choose $r$ elements from a set of $n$ elements is given by the combination formula $_{n}C_{r}=\frac{n!}{r!(n - r)!}$. We need to choose 5 women out of 30 women and 5 men out of 50 men.

Step2: Calculate number of ways to choose women

The number of ways to choose 5 women out of 30 is $_{30}C_{5}=\frac{30!}{5!(30 - 5)!}$.

Step3: Calculate number of ways to choose men

The number of ways to choose 5 men out of 50 is $_{50}C_{5}=\frac{50!}{5!(50 - 5)!}$.

Step4: Use multiplication - principle

By the multiplication - principle of counting (since the selection of women and men are independent events), the total number of ways to form a team of 5 women and 5 men is the product of the number of ways to choose women and the number of ways to choose men, which is $(_{30}C_{5})(_{50}C_{5})$.

Answer:

$(_{30}C_{5})(_{50}C_{5})$ (corresponding to the first option in the original problem where it seems the sub - script and super - script are a bit unclear in the image but should be $_{30}C_{5}$ and $_{50}C_{5}$ multiplied together)