Singers (E)nrique, (K)aty, (R)ihanna, and (B)runo are performing a charity concert for world hunger and their order of appearance will be chosen randomly. The outcomes in the sample space will be represented as strings of letters such as ERBK or KERB. Complete parts (a) through (c) below.
(b) Express the event "Katy and Bruno perform consecutively" as a set.
Answer
The event 'Katy and Bruno perform consecutively' can be expressed as the set \(\boxed{\{'EKBR', 'ERKB', 'ERBK', 'EBKR', 'KBER', 'KBRE', 'REKB', 'REBK', 'RKBE', 'RBKE', 'BKER', 'BKRE'\}}\).
Step 1 ：First, we need to generate all possible permutations of the four singers Enrique, Katy, Rihanna, and Bruno. These permutations are represented as strings of letters such as ERBK or KERB.
Step 2 ：The total permutations are: 'EKRB', 'EKBR', 'ERKB', 'ERBK', 'EBKR', 'EBRK', 'KERB', 'KEBR', 'KREB', 'KRBE', 'KBER', 'KBRE', 'REKB', 'REBK', 'RKEB', 'RKBE', 'RBEK', 'RBKE', 'BEKR', 'BERK', 'BKER', 'BKRE', 'BREK', 'BRKE'.
Step 3 ：Next, we need to filter out the permutations where Katy (K) and Bruno (B) perform consecutively.
Step 4 ：The permutations where Katy and Bruno perform consecutively are: 'EKBR', 'ERKB', 'ERBK', 'EBKR', 'KBER', 'KBRE', 'REKB', 'REBK', 'RKBE', 'RBKE', 'BKER', 'BKRE'.
Step 5 ：Finally, we express the event 'Katy and Bruno perform consecutively' as a set.
Step 6 ：The event 'Katy and Bruno perform consecutively' can be expressed as the set \(\boxed{\{'EKBR', 'ERKB', 'ERBK', 'EBKR', 'KBER', 'KBRE', 'REKB', 'REBK', 'RKBE', 'RBKE', 'BKER', 'BKRE'\}}\).
