please show all workings 2. A committee of 6 people, which must contain at least

Question

please show all workings

2. A committee of 6 people, which must contain at least 4 men and at least 1 woman, is to be chosen from 10 men and 9 women. (i) the number of possible committees that can be chosen. [3] Find (ii) Find the probability that one particular man, Anil, and one particular on the committee. [4] (iii) Find the number of possible committees that include either Anil or Stacy but not both. [3] (iv) The committee that is chosen consists of 4 men and 2 women. They line up randomly for Find the probability that the two women are not next to each other. [4]

Explanation / Answer

(i)

Total number of members : 10+9=19

There are 10 women and 9 men. Number of ways of selecting 4 men out of 10 men is

C(10,4) = 210

Number of ways of selecting 1 woman out of 9 women is

C(9,1) = 9

After selecting 4 men and 1 woman number of members remaining is 19 -5 = 14. Number of ways of selecting 1 member out of 14 is C(14,1)=14.

So number of ways of selecting 6 members such that at least 4 are men and 1 is woman is

210* 9 * 14 = 26,460

(ii)

Number of ways of selecting Anil and Stacy is 1. After that number of ways of selecting at least 3 men is

C(8,3) C(14,1) = 784

(iii)

Let us first find the number of committees that include only Anil not Stacy. After selecting Anil number of men remaining is 9 and since Stacy is not in the committee so number of women remaning is 8. So possible number of committees inlcuding Anil only is

C(9,3)C(8,1)C(13,1) = 8736

Likewise possible number of committees inlcuding Stacy only is

C(9,4)C(13,1) = 1638

SO number of poissble commitees including either Anil or Stacy but not both is 8736+1638 = 10374

(iv)

Total number of possible arrangements of 6 people is 6! = 720

Let women are together so number of ways arranging 5 persons in a line is 5! = 120 and two women can arrange themselves in 2!=2 ways. So possible line ups in which women are together is 120*2 = 240

So possible number of line ups in which women are not together is 720 - 240 = 480

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