(a) How many distinguishable ways can the letters of the word HULLABALOO be arranged in order?
(b)How many distinguishable ordering of the letters of HULLABALOO begin U and end with L?
(c)How many distinguishable orderings of the letters of HULLABALOO contain the two letters HU next to each other in order?

(a)

The number of distinguishable ways the letters of the word HULLABALOO can be arranged in order.

Given information:

The word HULLABALOO.

Concept used:

The straight lines can be formed by minimum 2 straight lines.

Calculation:

The given word is HULLABALOO

There are 10 positions and have to be filled with 1 H,1 U, 3L's, 2A's 1B and 2 O's.

Step 1: Choose 3 positions for 3L's.

This can be done in (103) ways.

Step 2: Choose 2 position for 2A's from the remaining 7 positions.

This can be done in (72) ways.

Step 3: Choose 2 position for 2 O's from the remaining 5 positions.

This can be done in (52) ways.

Steps 4: Choose 1 position for H from the remaining 3 positions.

This can be done in (31) ways.

Step 5: Choose 1 position for U from the remaining 2 positions

(b)

The number of distinguishable orderings of the letters of HULLABALOO begin with U and end with L.

(c)

The number of distinguishable orderings of the letters of HULLABALOO contain the two letters HU next to each other in order.