00:00In this question, we are asked how many 7-digit numbers can be found by using these digits.
00:05So we will solve simply, it does not matter whatever these 5 digits are, it matters that
00:10none of them is zero.
00:13If it was zero, then the formula and solving will be different.
00:17So first of all, we will take 7 places.
00:20Now each digit is repeated at least once.
00:24So 5 places are booked.
00:26Then we have to see, then 2 places remain empty.
00:29Then here we have 2 cases.
00:31First, one number is repeated 3 times and the others are simply.
00:36So how can we select the 3 places in which one number will be repeated 3 times.
00:42So we will select it by 7c3.
00:45Into how many choice sets that 3 times repeating number can have.
00:49It can be 1, 2, 3, 4, 5.
00:52So into 5, then in the remaining 4 digits, they can be arranged in 4 factorial ways.
00:58So it will be into 4 factorial.
01:00In the same way, 2nd catch will be, 1 digit is repeated 2 times and 2nd digit is also repeated
01:092 times.
01:11So first of all, let's say, we will select 2 places for the first digit to repeat.
01:16So 7c2.
01:18So 7c2.
01:18Into how much choices it takes?
01:20It can be 1, 2, 3, 4, 5.
01:22So into 5.
01:24Into, and now there remain 5 empty places.
01:28From which we have to select 2 more places for the second number to repeat.
01:32So 5c2.
01:34So 5c2.
01:35Why not 7c2?
01:36Because we cannot re-select the same 2 places which we have selected earlier.
01:42So 5c2.
01:43Into how many numbers left?
01:451 number was selected here.
01:47So we have now 4 numbers remaining.
01:50Into 4.
01:51Into 3 places remaining with 3 numbers.
01:541 was selected here.
01:551 was here.
01:56So 3.
01:57Which means into 3 factorial.
02:00Divided by 2.
02:01Why?
02:01Because suppose in first number we selected 1, second 2.
02:06Now we selected 2, 1.
02:08So it will be a repetition.
02:10Like 1, 1, 2, 2.
02:11And 2, 2, 1, 1.
02:13So it will be a 1, 1, 2, 2.
02:16It will be a repetition there.
02:19Because we have to select 2 numbers.
02:22But suppose in 7c2 the number we selected is 1.
02:27And here 2.
02:29But in the next case we have selected 2 in this.
02:32And 1 in this.
02:33So there is a repetition.
02:35So we divide it by 2.
02:37After we solve it.
02:407c3 into 5 into 4 factorial.
02:42Is equal to 4000.
02:44Yeah.
02:45This one.
02:477c3 into 5 into 4 factorial.
02:49Is equal to 4200.
02:51And this one 7c2 into 5 into 5.
02:54C2 into 4 into 3 factorial.
02:57Divided by 2.
02:58Is equal to 12600.
03:01If we add both.
03:02We get 16800.
03:04And here it is.
03:05And here it is.
03:05And that.
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