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Worked solution

Fill the most constrained digit position first

How many numbers between 3000 and 5000 can be formed using only the digits 1, 2, 3, 4 and 5 if (i) no digit is repeated, (ii) the digits can be repeated, (iii) no repetition is allowed and the numbers are even?

CJC · Lecture · Example 15

(i) Count four positions without repetition

The thousands digit must be 3 or 4: 2 choices. The hundreds, tens and units then have 4, 3 and 2 remaining choices. Hence 2×4×3×2=482\times4\times3\times2=48.

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(ii) Allow repetition in each later position

The thousands digit still has 2 choices. Each later position independently has 5 choices: 2×5×5×5=2502\times5\times5\times5=250.

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(iii) First digit 3: fill the units position next

The units digit is 2 or 4, giving 2 choices. Fill tens next with 3 remaining choices, then hundreds with 2. Thus 2×3×2=122\times3\times2=12. The displayed positional counts are fixed 3, then 2,3,2; the counting stages run units, tens, hundreds.

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(iii) First digit 4: the units digit is forced

Without repetition, the units digit must be 2. Tens has 3 choices and hundreds has 2. The displayed row is fixed 4, then 2 choices,3 choices, fixed 2. This contributes 1×3×2=61\times3\times2=6.

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(iii) Add the disjoint first-digit cases

There are 12+6=1812+6=18 even numbers. Follow the Teacher guidance: positions need not be filled left to right; usually begin with the most constrained position. Here fix the thousands digit to determine the available even units digits, then complete the middle positions.

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Four source rows in thousands, hundreds, tens, units order. Without repetition choices are 2,4,3,2; with repetition 2,5,5,5. For even numbers first digit 3 gives fixed 3,2 choices,3 choices,2 choices; first digit 4 gives fixed 4,2 choices,3 choices, fixed 2. In even cases choose units first, tens second, hundreds third after fixing the first digit.
Four source rows in thousands, hundreds, tens, units order. Without repetition choices are 2,4,3,2; with repetition 2,5,5,5. For even numbers first digit 3 gives fixed 3,2 choices,3 choices,2 choices; first digit 4 gives fixed 4,2 choices,3 choices, fixed 2. In even cases choose units first, tens second, hundreds third after fixing the first digit.
Four source rows in thousands, hundreds, tens, units order. Without repetition choices are 2,4,3,2; with repetition 2,5,5,5. For even numbers first digit 3 gives fixed 3,2 choices,3 choices,2 choices; first digit 4 gives fixed 4,2 choices,3 choices, fixed 2. In even cases choose units first, tens second, hundreds third after fixing the first digit.

Common mistakes

The first digit changes the units choices in part (iii); do not multiply a single units count across both first-digit cases. Neither endpoint 3000 nor 5000 can be made because 0 is unavailable.

Read the hint