Calculate the total number of distinct four-digit numbers that can be formed using the dig… — JAMB UTME

Calculate the total number of distinct four-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6,$ and $7$ without repetition, such that the resulting number is divisible by either $4$ or $5$ .

Answer: 320

Answer

The total number of such four-digit numbers is

320

The total number of valid four-digit numbers is found by adding the count of numbers divisible by

5

(

120

) to the count of numbers divisible by

4

(

200

). Since a number ending in

5

is odd, it cannot be divisible by

4

, making the two conditions mutually exclusive. Thus, the total count is

120 + 200 = 320

Step-by-Step Solution

Calculate the number of four-digit numbers divisible by

5

For a number to be divisible by

5

, its units digit must be

5

(since

0

is not available). There is

1

choice for the units digit. The remaining

3

positions are filled from the remaining

6

available digits in

^{6}P_{3} = 6 \times 5 \times 4 = 120

ways.

Divisibility by

5

requires the last digit to be

5

Calculate the number of four-digit numbers divisible by

4

A number is divisible by

4

if its last two digits form a multiple of

4

. Using distinct digits from

\{1, 2, 3, 4, 5, 6, 7\}

, the valid two-digit endings are

12, 16, 24, 32, 36, 52, 56, 64, 72,

and

76

(

10

valid pairs). For each pair, the first two positions are filled from the remaining

5

digits in

^{5}P_{2} = 5 \times 4 = 20

ways. Thus, total ways =

10 \times 20 = 200

Divisibility by

4

depends entirely on the last two digits.

Check for overlap (numbers divisible by both

4

and

5

A number divisible by

5

must end in

5

, which is an odd digit. All multiples of

4

must end in an even digit. Hence, no number is divisible by both

4

and

5

in this set. The overlap is

0

The two events are mutually exclusive.

Apply the addition principle of counting.

Total =

120 + 200 - 0 = 320

Add the counts of the two mutually exclusive sets.

Key Concept

Restricted Permutations and Mutually Exclusive Events

Estimated Time:1m 30s

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