The word PEPPER has 6 letters: three P's, two E's, and one R. How many distinct arrangements of these letters are there?
A720 (= 6!)
B60 (= 6! / (3! · 2! · 1!))
C120 (= 5!)
D360 (= 6! / 2!)
The multinomial coefficient corrects for repeated letters by dividing out all permutations among identical elements. With 3 P's, 2 E's, and 1 R, the formula is 6!/(3!·2!·1!) = 720/12 = 60. Option A (720) ignores the repeats entirely and overcounts — it assumes all 6 letters are distinct. Options C and D only account for one type of repetition instead of all.
Question 2 Multiple Choice
In the expansion of (a + b + c)⁴, what is the coefficient of the term a²bc?
A4
B6
C12
D24
The coefficient of a^k₁ b^k₂ c^k₃ in (a+b+c)ⁿ is the multinomial coefficient n!/(k₁!k₂!k₃!). Here n=4, k₁=2, k₂=1, k₃=1, so the coefficient is 4!/(2!·1!·1!) = 24/2 = 12. A common error is to use C(4,2) = 6 (option B), which only accounts for choosing the positions for a but ignores the subsequent distribution of b and c. The multinomial coefficient counts all the ways to assign the four 'slots' to a (2 slots), b (1 slot), and c (1 slot).
Question 3 True / False
The binomial theorem is a special case of the multinomial theorem that applies mainly when most exponents in the expansion are equal.
TTrue
FFalse
Answer: False
The binomial theorem is the m=2 special case of the multinomial theorem — it applies when there are exactly two terms being summed, regardless of the exponents. The multinomial theorem covers (x₁ + x₂ + ⋯ + xₘ)ⁿ for any number of terms m; setting m=2 gives exactly the binomial theorem with the familiar binomial coefficients C(n,k) = n!/(k!(n−k)!). Exponent equality is irrelevant.
Question 4 True / False
The number of distinct arrangements of a string of n letters, where letter i appears kᵢ times (with k₁ + k₂ + ⋯ + kₘ = n), is exactly the multinomial coefficient n!/(k₁!k₂!⋯kₘ!).
TTrue
FFalse
Answer: True
Correct. If all n letters were distinct, there would be n! arrangements. But among the arrangements, any permutation of the kᵢ identical copies of letter i produces the same word — so we are overcounting by kᵢ! for each group. Dividing by k₁!k₂!⋯kₘ! corrects all overcounting simultaneously, giving n!/(k₁!⋯kₘ!). This is the core combinatorial meaning of the multinomial coefficient.
Question 5 Short Answer
Why does the multinomial coefficient formula divide by each kᵢ! separately rather than, say, by (k₁ + k₂ + ⋯ + kₘ)! or by the product k₁ · k₂ · ⋯ · kₘ? What does each factorial represent?
Think about your answer, then reveal below.
Model answer: Each kᵢ! accounts for the kᵢ! ways of permuting the identical copies of item i among themselves — permutations that produce the same arrangement and must not be counted separately. Since the groups are independent, the total overcounting is the product of all the individual kᵢ! values, so we divide by each factorial independently. Dividing by the sum (k₁+⋯+kₘ)! = n! would over-correct, and dividing by the product of the raw values kᵢ wouldn't fully remove all redundant permutations.
The factorial kᵢ! appears because the kᵢ identical copies of item i can be rearranged among themselves in kᵢ! ways, all yielding the same outcome. Since these groups of identical objects are independent of each other, the overcounting factors multiply: total overcounting = k₁! × k₂! × ⋯ × kₘ!. Dividing n! by this product gives the exact count of truly distinct arrangements.