A project team of 5 members is to be formed from 6 senior engineers and 4 junior engineers… — JAMB UTME

A project team of $5$ members is to be formed from $6$ senior engineers and $4$ junior engineers. If the team must contain strictly more senior engineers than junior engineers, in how many different ways can the team be selected?

A
$180$
B
$252$
$186$ Cevap
D
$3600$

Cevap

186

To form a

5

-member team where senior engineers outnumber junior engineers, we consider three mutually exclusive cases: (5 senior, 0 junior), (4 senior, 1 junior), and (3 senior, 2 junior). Evaluating

^6C_5 \times ^4C_0 = 6

^6C_4 \times ^4C_1 = 60

, and

^6C_3 \times ^4C_2 = 120

, and adding them together yields

6 + 60 + 120 = 186

total ways.

Adım Adım Çözüm

Identify the conditions and valid cases for team composition

The team must have

5

members. The number of senior engineers must be strictly greater than junior engineers. The valid combinations of (senior, junior) engineers are:

(5, 0)

(4, 1)

, and

(3, 2)

Any other breakdown (such as 2 senior and 3 junior) would result in senior engineers not outnumbering junior engineers.

Calculate the combinations for Case 1 (

5

senior,

0

junior)

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

Choosing

5

out of

6

senior engineers and

0

out of

4

junior engineers.

Calculate the combinations for Case 2 (

4

senior,

1

junior)

^6C_4 \times ^4C_1 = 15 \times 4 = 60

Choosing

4

out of

6

senior engineers and

1

out of

4

junior engineers.

Calculate the combinations for Case 3 (

3

senior,

2

junior)

^6C_3 \times ^4C_2 = 20 \times 6 = 120

Choosing

3

out of

6

senior engineers and

2

out of

4

junior engineers.

Sum the combinations across all mutually exclusive cases

Total ways =

6 + 60 + 120 = 186

By the addition principle of counting, the total number of ways is the sum of ways from each disjoint case.

Anahtar Kavram

Conditional combinations using the formula

^nC_r = \frac{n!}{r!(n-r)!}

and the addition principle for disjoint cases.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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