How many numbers greater than $3400$ can be formed using first five natural numbers?
Without and with repetition.
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2With repetition would give a fair number I think - is there a restriction on number of digits perhaps? – Mark Bennet Mar 12 '21 at 12:18
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2@sdpmaths Where are you stuck? – Ishraaq Parvez Mar 12 '21 at 12:23
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@Ishraaq parvez I have been calculating 4 digits and 5 digits numbers separately and they are 50 and 12 respectively. – SandyM Mar 12 '21 at 12:25
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What does "formed" mean? Adding, multiplying, subtracting? Just putting the digits together in base 10? – aschepler Mar 12 '21 at 12:26
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@aschepler formed means make a number by using 1, 2, 3, 4 and 5 digits – SandyM Mar 12 '21 at 12:29
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@sdpmaths You should edit the question to show your calculations and explain your thinking, so an answer can give more specific feedback. – aschepler Mar 12 '21 at 12:37
1 Answers
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Let's talk without repetition first.
All 5 digit numbers would give $^5P\ _5=5!$
And 3, 2, and 1 digit numbers are out of the question.
And in 4 digit numbers, keeping the first digit as 4 or 5 would straight-up give an acceptable number. Hence, we have $2\times^4P\ _3$(fixing the first digit and arranging the remaining 3 digits using 4 available digits).
If the first digit is 3 then the second digit must be 4 or 5. So there goes another $2\times^3P\ _2$.
The first digit cannot be 1 or 2.
So we have $$ 5P\ _5 + 2\times ^4P\ _3 + 2\times^3P\ _2 $$ $$ =120 + 2\times 24 + 2\times 6 $$ $$ =120 + 48 + 12 $$ $$ =180 $$
So 180 such numbers are possible without repetition.
As for with repetition, infinite numbers are possible. To give you an idea, the following are some such numbers. 11111,111111,1111111,11111111,...
