PLZ Help me with the following questions!
very urgent!
The marketing department of a company would like to introduce 12 monthly special products in the coming 12 months. If these monthly special products are selected randomly from 24 products, find the probability that two specific products are the monthly special products in two consecutive months.
To find the probability that two specific products are selected as the monthly special products in two consecutive months, we can break down the problem into smaller steps.
Step 1: Determine the total number of possible outcomes.
Since there are 24 products to choose from and the marketing department selects 12 monthly special products, we can use the concept of combinations to find the total number of possible outcomes. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items being chosen. In this case, we have 24 products and we are choosing 12, so the total number of possible outcomes is 24C12 = 24! / (12!(24-12)!) = (24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13) / (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1).
Step 2: Determine the number of favorable outcomes.
To have two specific products as the monthly special products in two consecutive months, we need to choose those two products and then arrange them consecutively. So, we first choose 2 products from the 24 available. The number of ways to choose 2 products from 24 is 24C2 = 24! / (2!(24-2)!) = (24 x 23) / (2 x 1). Once we have chosen the 2 products, we can arrange them in two consecutive months, which is 2! = 2 x 1.
Step 3: Calculate the probability.
Now, we can calculate the probability by dividing the number of favorable outcomes (Step 2) by the total number of possible outcomes (Step 1).
Probability = Number of favorable outcomes / Total number of possible outcomes
= (Number of ways to choose 2 products from 24) x (Number of ways to arrange the 2 products) / (Total number of ways to choose 12 products from 24)
= ((24 x 23) / (2 x 1)) x (2 x 1) / ((24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13) / (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1))
= (24 x 23 x 2 x 1) / (24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13)
Simplifying further, we can cancel out some terms:
Probability = (2 x 1) / (22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13)
Finally, calculate the Probability using a calculator:
Probability ≈ 0.00000001190476
Therefore, the probability that two specific products are the monthly special products in two consecutive months is approximately 0.00000001190476 or 1.19 x 10^-11.