A committee of 5 members is to be selected from 6 doctors and 4 nurses. In how many ways c… — JAMB UTME

A committee of $5$ members is to be selected from $6$ doctors and $4$ nurses. In how many ways can this committee be formed if it must include at least $3$ doctors?

186Cevap
B
120
C
720
D
3600

Cevap

186 ways

The required committee must contain at least 3 doctors out of 5 total members. The three possible scenarios are: 3 doctors and 2 nurses (

^6C_3 \times ^4C_2 = 120

), 4 doctors and 1 nurse (

^6C_4 \times ^4C_1 = 60

), and 5 doctors and 0 nurses (

^6C_5 \times ^4C_0 = 6

). Summing these gives

120 + 60 + 6 = 186

Adım Adım Çözüm

Identify the possible valid committee compositions given the condition 'at least 3 doctors'.

Three mutually exclusive cases exist for a 5-member committee: (3 doctors, 2 nurses), (4 doctors, 1 nurse), or (5 doctors, 0 nurses).

The total size of the committee is 5, so selecting more doctors reduces the required number of nurses.

Calculate the combinations for Case 1: 3 doctors and 2 nurses.

^6C_3 \times ^4C_2 = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} \times \frac{4 \times 3}{2 \times 1} = 20 \times 6 = 120

Order of selection within each group does not matter.

Calculate the combinations for Case 2: 4 doctors and 1 nurse.

^6C_4 \times ^4C_1 = \frac{6 \times 5}{2 \times 1} \times 4 = 15 \times 4 = 60

Selecting 4 doctors out of 6 is equivalent to choosing which 2 doctors to exclude.

Calculate the combinations for Case 3: 5 doctors and 0 nurses.

^6C_5 \times ^4C_0 = 6 \times 1 = 6

Choosing 5 doctors out of 6 yields 6 possibilities, and choosing 0 nurses yields 1.

Sum the number of ways from all valid cases.

120 + 60 + 6 = 186

By the addition principle of counting, the total ways to form the committee is the sum of ways across mutually exclusive cases.

Anahtar Kavram

Combinations with constraints (addition and multiplication principles)

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