in how many ways could a band of singers and 5 drummers be selected from 4 singers and 6 drummers.of a band of 7 members are to be selected from a team of 4 singers and 6 drummers.find the probability of including 2 singers
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To find the number of ways a band of singers and drummers can be selected, we use the concept of combinations.
1. Finding the number of ways to select a band:
To calculate the number of ways, we need to find the combinations of singers and drummers together, using the formula for combinations:
nCr = n! / (r!(n-r)!)
where n is the total number of people to choose from (in this case, the total number of singers and drummers), and r is the number of members to be selected (in this case, the number of singers and drummers in the band).
In this case, we have 4 singers and 6 drummers. Let's assume we want to select a band of 7 members (including both singers and drummers).
So, the total number of ways to select a band from this group is given by:
(4+6)C7 = 10C7 = 10! / (7!(10-7)!) = 10! / (7!3!) = (10 * 9 * 8) / (3 * 2 * 1) = 120 ways
Therefore, there are 120 different ways to select a band of 7 members from the group of 4 singers and 6 drummers.
2. Finding the probability of including 2 singers:
To calculate the probability, we need to determine the favorable outcomes and divide it by the total number of possible outcomes.
The favorable outcomes in this case would be the number of ways to select a band that includes 2 singers. To calculate this, we choose 2 singers from the group of 4 singers, and then select the remaining 5 members from the group of drummers:
(4C2) * (6C5) = (4! / (2!(4-2)!)) * (6! / (5!(6-5)!)) = (4! / (2!2!)) * (6! / (5!1!)) = (4! * 6) / (2!2!) = 24
Therefore, there are 24 favorable outcomes.
The total number of possible outcomes is the same as the total number of ways to select a band, which is 120 (calculated earlier).
So, the probability of including 2 singers in a band of 7 members is:
24 / 120 = 1/5 = 0.2 (or 20%)
Hence, the probability of including 2 singers in a band of 7 members is 0.2 or 20%.