you have 14 colors, and you will make 7 colored letters. Notice that the letters of consider are all different, so the order of your coloring is important (in other words, writing CONSIDER with a red "C" and a green "O" , etc, is different than writing CONSIDER with a green "C" and a red "O")
Since order is important, the computation is 14P7 (14 permute 7). The equation for this 14! / (14 - 7)! = 14! / 7! = 14*13*12*11*10*9*8 = 17297280
question 2.
for the first part, notice you have a total of 5 digits to choose from and must make a 5 digit number. Consider your number as 5 "slots" each one can have one of the numbers 2,3,4,5,6.
If repetition is allowed, then there are 5 possibilities for the first slot (the first number can be either 2, 3, 4, 5, or 6), also 5 possibilities for the second, third, fourth, and 5th slots. Notice that the choice of digit for any slot has no effect on what the other digits can be (for part a only). Thus, by the multiplication rule you have 5^5 = 3125 numbers
for b, repetition is not allowed, so each choice of digit for one slot affects the others. So for your first digit you have 5 choices. For your second digit, you can't choose the same number as you did for the first one so you have 4 choices. For your third digit you have 3 choices, then two, and finally for the final digit you only have one number left and must choose it. So the number of combinations here is
5*4*3*2*1 = 120
notice this is equal to 5P5 (5 permute 5) as you have 5 digits that you can order any way you please to make a 5 digit number.
Math is fun.
This post was edited by sumi9lives on Jan 26 2015 12:25am