First slide

Combinations

Question

The number of integers between 1 and 1000000 that have the sum of the digits 18, is

Moderate

A

25927
B

25827
C

24927
D

None of these

Solution

Any number between 1 and 1000000 must be of less than seven digits. Therefore, it must be of the form a₁ a₂ a₃ a₄ a₅ a₆
where a₁, a₂, a₃, a₄, a₅, a₆ ∈ {0, 1, 2, …, 9}
According to question, sum of the digits = 18
Thus, a₁ + a₂ + a₃ + a₄ + a₅ + a₆ = 18
where 0 ≤ a_i ≤ 9, i = 1, 2, 3, …, 9.
Required number
$\begin{array}{l} = coefficient of x^{18} in {(1 + x + x^{2} + \dots + x^{9})}^{6} \\ = coefficient of x^{18} in {(\frac{1 - x^{10}}{1 - x})}^{6} \\ \begin{array}{l} = coefficient of x^{18} in [{(1 - x^{10})}^{6} (1 - x)^{- 6}] \\ = coefficient of x^{18} in [(1 -^{6} C_{1} x^{10}) (1 - x)^{- 6}] \end{array} \end{array}$
[leaving terms containing powers of x greater than 18]
$= coefficient of x^{18} in (1 - x)^{- 6} -^{6} C_{1} . coefficient of x^{8} in (1 - x)^{- 6} \begin{array}{l} =^{6 + 18 - 1} C_{18} - 6 \cdot^{6 + 8 - 1} C_{18} =^{23} C_{5} - 6 \cdot^{13} C_{8} \\ = \frac{23 \cdot 22 \cdot 21 \cdot 20 \cdot 19}{120} - 6 \cdot \frac{13 \cdot 12 \cdot 11 \cdot 10 \cdot 9}{120} \\ = 33649 - 7722 = 25927 \end{array}$

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