Counting principles; binomial theorem extension practice questions.

Four different mathematics books and three different physics books are placed in a row on a shelf. Determine the number of arrangements in which the three physics books are all next to each other.

• 5040
• 720
• 144
• 240

Answer: 720

Treat the physics books as one block: 5 units arrange in 5! ways, and the block arranges internally in 3! ways, giving 120 × 6 = 720.

The first three terms of the expansion of (1 + x)^(1/2) in ascending powers of x are

• 1 + x/2 + x²/8
• 1 − x/2 − x²/8
• 1 + x/2 − x²/4
• 1 + x/2 − x²/8

Answer: 1 + x/2 − x²/8

With n = 1/2 the x² coefficient is n(n − 1)/2! = (1/2)(−1/2)/2 = −1/8, so the expansion begins 1 + x/2 − x²/8. Forgetting the minus sign in (n − 1) is the classic error.

State the set of values of x for which the infinite binomial expansion of (1 − 2x)⁻¹ is valid.

• |x| < 1/2
• |x| < 1
• |x| < 2
• all real x

Answer: |x| < 1/2

The expansion of (1 + u)ⁿ for non-natural n converges for |u| < 1; here u = −2x, so |−2x| < 1 gives |x| < 1/2.

How many four-digit numbers with all digits different can be formed using only the digits 1 to 7?

• 2401
• 35
• 840
• 5040

Answer: 840

Order matters and repetition is not allowed, so the count is 7 × 6 × 5 × 4 = 840. Using 7⁴ = 2401 ignores the no-repetition condition.

Determine the coefficient of x² in the expansion of (1 + x)⁻³.

• −6
• 6
• 12
• 3

Answer: 6

The x² coefficient is n(n − 1)/2! with n = −3, i.e. (−3)(−4)/2 = 6. The two negative factors multiply to give a positive result.

Six people are to be seated at a circular table. If two seatings are considered the same when one is a rotation of the other, the number of distinct seatings is

• 720
• 240
• 24
• 120

Answer: 120

Fix one person to remove the rotational symmetry; the remaining five can then be arranged in 5! = 120 ways.