A music festival coordinator needs to select 4 bands to perform from a pool of 8 available… — JAMB UTME

A music festival coordinator needs to select $4$ bands to perform from a pool of $8$ available bands. If $2$ specific bands insist on either both being selected or neither being selected, in how many different ways can the $4$ bands be chosen?

$30$ Cevap
B
$60$
C
$70$
D
$390$

Cevap

The total number of ways to choose the bands under the given condition is

30

To satisfy the condition that the two specific bands are either both chosen or neither chosen, we evaluate two distinct cases. Case 1 (both chosen) requires choosing

2

additional bands from the remaining

6

, yielding

{}^6C_2 = 15

ways. Case 2 (neither chosen) requires choosing all

4

bands from the remaining

6

, yielding

{}^6C_4 = 15

ways. Adding both cases gives

15 + 15 = 30

ways.

Adım Adım Çözüm

Analyze Case 1: Both specific bands are selected.

If both specific bands are included, we only need to select

2

more bands from the remaining

6

bands. The number of ways is

{}^6C_2 = \frac{6 \times 5}{2 \times 1} = 15

Since order does not matter in forming a group of performers, we use combinations.

Analyze Case 2: Neither of the specific bands is selected.

If neither of the

2

specific bands is chosen, all

4

bands must be selected from the remaining

6

bands. The number of ways is

{}^6C_4 = {}^6C_2 = 15

Excluding the

2

specific bands leaves

6

candidate bands to choose

4

from.

Sum the mutually exclusive cases.

\text{Total ways} = 15 + 15 = 30.

The two scenarios are disjoint, so the addition principle applies.

Anahtar Kavram

Combinations with conditional restrictions

Tahmini Süre:1m 30s

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