Question

How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4.

Answer 1

Hint: In this question, we need to find the number greater than 1000000 using the digits 1, 2, 0, 2, 4, 2, 4. For this, we will first find the total number that can be formed using 7 digits. After that, we will divide the repeated number by the total number. At last, we will subtract the numbers for which 0 comes at ten lakh from the found number to get our final answer. Total number of arrangements of n items is given by n! If p of one kind are repeated, q of other types are repeated in those n items then the number of arrangements are $\dfrac{n!}{p!q!}$.

Complete step by step answer:
Here, we are given the digits as 1, 2, 0, 2, 4, 2, 4. We have to form numbers greater than 1000000. So we have to use all the given seven digits. Now we know that, number of arrangements using n items is given by n! So for 7 digits, the number of arrangements could be 7!. But as we can see, 2's and 4's are repeated i.e. 2 comes three times and 4 comes 2 times, so we have to reduce our numbers. We know that, for n items with p repeated items of one kind and q repeated items of other kinds, the number of arrangements are $\dfrac{n!}{p!q!}$. So here n = 7, p = 3, q = 2.
Number of arrangement $\Rightarrow \dfrac{7!}{3!2!}=\dfrac{7\times 6\times 5\times 4\times 3!}{3!\times 2}=\dfrac{7\times 6\times 5\times 4}{2}=7\times 6\times 5\times 2=420$.
Now since 0 was one of our digits, we must have counted numbers with 0 in ten lakhs place, put it will make our number less than 1000000. So we need to remove those numbers from the total count found earlier.
Keeping 0 fixed at ten lakhs place, counting six digit numbers with 1, 2, 2, 4, 2, 4. Number of arrangements using 6 digits is 6!
Repeated digits are 3 two's and 2 fours. So, number of arrangement will be $\Rightarrow \dfrac{6!}{3!2!}=\dfrac{6\times 5\times 4\times 3!}{3!\times 2}=\dfrac{6\times 5\times 4}{2}=6\times 5\times 2=60$.
So we need to remove these 60 numbers which have 0 in ten lakhs place.
Hence we get required numbers as $420-60=360$.

So, the correct answer is “360”.

Note: Students often forget to remove the number with 0 in the ten lakh place. Take care in calculation. Make sure to divide the repeated numbers. Students can make the mistake of dividing n! by p and q only. But we need to divide n! by p! and q!.