Question

Using all or some of the digits 2, 3, 4 and 5, Edward writes numbers greater than 500 without repeating digits. For example, he might write 543.
How many numbers does Edward write?

Answer 1

Given: the digits 2, 3, 4 and 5

To find: how many numbers does Edward write?

Solution:

To form numbers greater than 500, in each number, we must take the digit 5 in an appropriate place because 2, 3, 4 < 5.

• The possible outcomes are:
• (2, 3, 5) = 532, 523
• (2, 4, 5) = 542, 524
• (3, 4, 5) = 543, 534
• (2, 3, 4, 5) = 2345, 2435, 2354, 2453, 3245, 3425, 3254, 3452, 4325, 4235, 4352, 4253, 5234, 5234, 5243, 5324, 5342, 5432 and 5423.

Thus total number of requires outcomes is

= 2 + 2 + 2 + 20 = 26

Answer: Edward writes 26 numbers.

Answer 2

Given:

We are given some digits - 2, 3, 4, and 5.

To find:

We need to find out total numbers greater than 500 without repeating digits using the given digits.

Solution:

• Using 2, 3, 4, and 5, we need to create numbers greater than 500 without repeating the digits. So to find such numbers we use the concept of permutation and combination.
• We will take two cases as numbers greater than 500 can be a 3-digit number or a 4-digit number since we are given only 4 digits to use i.e. 2, 3, 4, and 5.

Case 1:

In this case, we will find the 3-digit numbers greater than 500 formed from the given digits i.e. 2, 3, 4, and 5.

As we know that if we need to find numbers greater than 500 from 2, 3, 4, and 5, we should take the first digit of the 3-digit number as 5.

⇒ The 3-digit numbers greater than 500 formed from the given digits i.e. 2, 3, 4, and 5 should look like this 5 _ _.

Now find the number of ways in which these remaining two places can be filled.

Since there are 3 options for the tens place i.e. 2, 3, and 4 (5 is already there in the number and it cant be repeated due to the given condition).

∴ Now only 2 options will be left for the ones place as one number from 2, 3, and 4 have been filled by tens place.

∴ Total choices for the ones and tens place = 2*3 options = 6 options.

Case 2:

In this case, we will find the 4-digit numbers greater than 500 formed from the given digits i.e. 2, 3, 4, and 5.

Since all 4-digit numbers are greater than 500 because it is a 3-digit number.

⇒ All 4-digit numbers formed from digits 2, 3, 4, and 5 are greater than 500.

⇒Finding all 4-digit numbers formed from digits 2, 3, 4, and 5:

Let the 4-digit number - _ _ _ _

It has 4 places to fill.

• The thousandth place has 4 options - 2, 3, 4, and 5.

• The hundredth place has 3 options - one less than the thousandth place since one digit would have been filled by the thousandth place.

• The tens place has 2 options - one less than the hundredth place.

• The ones place has 1 option - one less than tens place.

∴ Total 4-digit numbers formed from digits 2, 3, 4, and 5 are greater than 500 are 4×3×2×1 options = 24 options.

Answer:

Combining both the cases - a total of (6+24) options.

Total numbers greater than 500 without repeating digits using the given digits i.e. 2, 3, 4, and 5 are 30.