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Question Number 174855 by byaw last updated on 12/Aug/22

Find the number of ways   a committee of 4 people can be  chosen from a group of 5 men  and 7 women when it contains  people of both sexes and  there are at least as many  women as men.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\: \\ $$$$\mathrm{a}\:\mathrm{committee}\:\mathrm{of}\:\mathrm{4}\:\mathrm{people}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{chosen}\:\mathrm{from}\:\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\mathrm{5}\:\mathrm{men} \\ $$$$\mathrm{and}\:\mathrm{7}\:\mathrm{women}\:\mathrm{when}\:\mathrm{it}\:\mathrm{contains} \\ $$$$\mathrm{people}\:\mathrm{of}\:\mathrm{both}\:\mathrm{sexes}\:\mathrm{and} \\ $$$$\mathrm{there}\:\mathrm{are}\:\mathrm{at}\:\mathrm{least}\:\mathrm{as}\:\mathrm{many} \\ $$$$\mathrm{women}\:\mathrm{as}\:\mathrm{men}. \\ $$

Answered by Kallu last updated on 12/Aug/22

  case 1:   2 men and 2 women          no. of ways=^5_C_2    ×^7_C_2    = 210    case 2:  1 man and 3 women          no. of ways=^5_C_1    ×^7_C_3    = 175    Therefore total ways = 385

$$ \\ $$$${case}\:\mathrm{1}:\:\:\:\mathrm{2}\:{men}\:{and}\:\mathrm{2}\:{women} \\ $$$$\:\:\:\:\:\:\:\:{no}.\:{of}\:{ways}=\:^{\mathrm{5}_{{C}_{\mathrm{2}} } } \:×\:^{\mathrm{7}_{{C}_{\mathrm{2}} } } \:=\:\mathrm{210} \\ $$$$ \\ $$$${case}\:\mathrm{2}:\:\:\mathrm{1}\:{man}\:{and}\:\mathrm{3}\:{women} \\ $$$$\:\:\:\:\:\:\:\:{no}.\:{of}\:{ways}=\:^{\mathrm{5}_{{C}_{\mathrm{1}} } } \:×\:^{\mathrm{7}_{{C}_{\mathrm{3}} } } \:=\:\mathrm{175} \\ $$$$ \\ $$$${Therefore}\:{total}\:{ways}\:=\:\mathrm{385}\: \\ $$$$ \\ $$

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