Four band members, Harry, Isabel, Jose, and Karen, are all boarding a bus to attend a concert. In each arrangement below, the first initial of each person’s name represents that person’s position as he or she boards the bus: first, second, third, and fourth.

Question

If we know that Karen boards the bus first, which shows the possible outcomes for this event?

KHIJ
KHIJ, KHJI
KHIJ, KHJI, KJHI, KJIH
KHIJ, KHJI, KJHI, KJIH, KIHJ, KIJH

Answer 1

Given that Karen (K) boards the bus first, we need to consider the arrangements of the remaining band members: Harry (H), Isabel (I), and Jose (J).

Since K is fixed in the first position, we only need to arrange H, I, and J in the last three positions.

The possible arrangements for H, I, and J are:

1. H I J
2. H J I
3. I H J
4. I J H
5. J H I
6. J I H

Now, we'll append K to the start of each of these arrangements:

1. K H I J
2. K H J I
3. K I H J
4. K I J H
5. K J H I
6. K J I H

So, the complete list of arrangements with K in the first position is:

• KHIJ
• KHJI
• KIHJ
• KIJH
• KJHI
• KJIH

Thus, the correct outcome set is:

KHIJ, KHJI, KIHJ, KIJH, KJHI, KJIH.

Looking at the options provided in the question:

The last choice, which includes KHIJ, KHJI, KJHI, KJIH, would be incomplete but represents certain correct arrangements. However, it misses KIHJ and KIJH.

Therefore, none of the provided outcomes show all possible arrangements precisely. But the option that broadly shows different arrangements accurately would be:

KHIJ, KHJI, KJHI, KJIH.

However, for total completeness, all should include KIHJ and KIJH. Thus, the best answer among those provided is not fully appropriate as it’s incomplete.