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From a group of 6 men and 4 women a committee of 4 persons is to be formed. 13. In how many different ways can it be done so that the committee has atleast one woman? (A) 210 (B) 225 (C) 195 (D) 185 14. In How many different ways can it be done so that the committee has at least 2 men? (A) 210 (B) 225 (C) 195 (D) 185?
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From a group of 6 men and 4 women a committee of 4 persons is to be fo...
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13. At least one woman in the committee:
To find the number of ways in which a committee of 4 persons can be formed with at least one woman, we can use the complement rule. That is, we first find the total number of committees that can be formed from 10 people (6 men and 4 women) without any restrictions, and then subtract the number of committees with no women.
Total number of committees = $^{10}C_4$ = 210
Number of committees with no women = $^{6}C_4$ = 15
Therefore, the number of committees with at least one woman = Total number of committees - Number of committees with no women
= 210 - 15
= 195
Hence, the answer is option (C) 195.
14. At least two men in the committee:
To find the number of ways in which a committee of 4 persons can be formed with at least two men, we can use the combination of cases.
Case 1: 2 men and 2 women
Number of ways to select 2 men from 6 men = $^6C_2$ = 15
Number of ways to select 2 women from 4 women = $^4C_2$ = 6
Number of committees with 2 men and 2 women = 15 x 6 = 90
Case 2: 3 men and 1 woman
Number of ways to select 3 men from 6 men = $^6C_3$ = 20
Number of ways to select 1 woman from 4 women = $^4C_1$ = 4
Number of committees with 3 men and 1 woman = 20 x 4 = 80
Case 3: 4 men and 0 women
Number of ways to select 4 men from 6 men = $^6C_4$ = 15
Number of committees with 4 men and 0 women = 15 x 1 = 15
Therefore, the total number of committees with at least two men = Number of committees with 2 men and 2 women + Number of committees with 3 men and 1 woman + Number of committees with 4 men and 0 women
= 90 + 80 + 15
= 185
Hence, the answer is option (D) 185.
Note: The sum of the number of committees in each case must be equal to the total number of committees formed.
13. At least one woman in the committee:
To find the number of ways in which a committee of 4 persons can be formed with at least one woman, we can use the complement rule. That is, we first find the total number of committees that can be formed from 10 people (6 men and 4 women) without any restrictions, and then subtract the number of committees with no women.
Total number of committees = $^{10}C_4$ = 210
Number of committees with no women = $^{6}C_4$ = 15
Therefore, the number of committees with at least one woman = Total number of committees - Number of committees with no women
= 210 - 15
= 195
Hence, the answer is option (C) 195.
14. At least two men in the committee:
To find the number of ways in which a committee of 4 persons can be formed with at least two men, we can use the combination of cases.
Case 1: 2 men and 2 women
Number of ways to select 2 men from 6 men = $^6C_2$ = 15
Number of ways to select 2 women from 4 women = $^4C_2$ = 6
Number of committees with 2 men and 2 women = 15 x 6 = 90
Case 2: 3 men and 1 woman
Number of ways to select 3 men from 6 men = $^6C_3$ = 20
Number of ways to select 1 woman from 4 women = $^4C_1$ = 4
Number of committees with 3 men and 1 woman = 20 x 4 = 80
Case 3: 4 men and 0 women
Number of ways to select 4 men from 6 men = $^6C_4$ = 15
Number of committees with 4 men and 0 women = 15 x 1 = 15
Therefore, the total number of committees with at least two men = Number of committees with 2 men and 2 women + Number of committees with 3 men and 1 woman + Number of committees with 4 men and 0 women
= 90 + 80 + 15
= 185
Hence, the answer is option (D) 185.
Note: The sum of the number of committees in each case must be equal to the total number of committees formed.
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From a group of 6 men and 4 women a committee of 4 persons is to be formed. 13. In how many different ways can it be done so that the committee has atleast one woman? (A) 210 (B) 225 (C) 195 (D) 185 14. In How many different ways can it be done so that the committee has at least 2 men? (A) 210 (B) 225 (C) 195 (D) 185?
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