Solved: Consider the collection of six-digit positive integers that can be created from... [IB Mathematics Analysis and Approaches (AA)]

Question

HLPaper 2

Consider the collection of six-digit positive integers that can be created from the digits $0, 1, 2, 3, 4, 5, 6, 7, 8,$ and $9$ . Determine the total number of six-digit positive integers that can be constructed such that

The digits are unique.

[2]

Verified

Solution

${9}\times{9}\times{8}\times{7}\times{6}\times{5}$ (M1)

${=136080}={9}\times{\frac{9!}{4!}}$ A1

Note: Award M1A0 for ${10}\times{9}\times{8}\times{7}\times{6}\times{5}$ = ${P_6} = {151200} = {\frac{10!}{4!}}$

Note: Award M1A0 for ${P_6} = {9}$ = ${60480}$

[2 marks]

The digits are unique and are in ascending order.

[2]

Verified

Solution

METHOD 1

EITHER every unordered subset of $6$ digits from the set of $9$ non-zero digits can be arranged in exactly one way into a $6$ -digit number with the digits in ascending order. A1

OR $\binom{6}{9} \times 1$ A1

THEN $=84$ A1

METHOD 2

EITHER removes $3$ digits from the set of $9$ non-zero digits and these $6$ remaining digits can be arranged in exactly one way into a $6$ -digit number with the digits in ascending order. A1

OR $\binom{3}{9} \times 1$ A1

THEN $=84$ A1

[2 marks]

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