Question

3. An urn contains five white balls numbered from 1 to 5, five red balls numbered from 1 to 5 and five blue balls numbered from 1 to 5. For each of the following questions, please give your answer first in the form that reflects your counting process, and then simplify that to a number. You must include the recipes. No other explanation needed. (a) In how many ways can we choose 4 balls from the urn? (b) in how many ways can we choose 4 balls from the urn so that there is at least one red ball among the chosen balls? (c) In how many ways can we choose 4 balls from the urn so that there is at least one red ball and at least one blue ball among the chosen balls? (d) In how many ways can we choose 4 balls from the urn so that the numbers on the chosen balls are distinct?

Answer 1

a) There are a total of 15 balls in the urn now we have to choose 4 balls from the urn. Thus the number of possible ways is:

$\small \binom{15}4=1365$

b) Here we will first calculate the number of ways in which there are no red balls chosen. Thus we have to choose 4 balls from the remaining 10 balls of remaining colors. Thus the number of ways are:

$\small \binom{10}4=210$. Hence the number of ways in which at least one red ball is chosen is 1365-210 = 1155.

c) Here we want to find the number of ways where there is at least one red ball and at least one blue ball. Instead, we will calculate the number of ways in which there is no red ball or no blue ball.

Number of ways in which there is no red ball or no blue ball is chosen = no red ball is chosen + no blue ball is chosen - no blue and no red ball is chosen = $\small \binom{10}4+\binom{10}4+\binom{5}4=210+210+5=425$

Thus the number of ways in which there is at least one red ball and at least one blue ball = Total ways - 425 = 940

d) Here first, we will choose e the 4 numbers out of 5 possible to be chosen, this can be done in 5 ways. Now for each number, there are 3 options thus the total number of ways= 5* 3⁴= 405