How many five-digit numbers have distinct digits which are decreasing from left to right? (For example, 96531 is such a number.)
A committee of 4 is to be chosen from a group of students. If the number of students in the group increases by 1, the number of different committees doubles. How many students are in the group?
Thanks!
+2864 I don't have time so I'm going to answer the last one
If you are to choose 4 out of x students it will be \(\frac{x!}{4!(x-4)!}=y\)
Then if the number of students in the group increases by 1, the committees double.
So \(\frac{(x+1)!}{4!((x+1)-4)!}=2y\)
So we have two equations with 2 variables.
You can try and solve now, (try substituting)
How many five-digit numbers have distinct digits which are decreasing from left to right? (For example, 96531 is such a number.)
1- ABCDE. I will give this a shot.
2 - For A, you have 9 choices.
3 - For B, you still have 9 choices out of 10, because we are excluding A
4 - For C, you have 8 choices because we are excluding A and B
5 - For D, you have 7 choices because we are excluding A, B, and C.
6 - For E, you have 6 choices because we are excluding A, B, C and D.
7 - Then we have: 9 x 9 x 8 x 7 x 6 =27,216 - numbers.
8 - The biggest such number would be: 98765.
9 - The smallest such number would be: 43210
10 - The difference:98765 - 43210 =55,555 from which 27,216 would be descending numbers.
Note: somebody should check this. Thanks.
