A magic wish-granting rectangle belt always shrinks to 1/2 its length and 1/3 its width whenever its owner makes a wish. After three wishes, the surface area of the belt's front side was 4 cm2. What was the original length, if the original width was 9 cm

(a) (8%) compute the probability of an even integer among the 100 integers 1! , 2! , 3! , until 100! (here n! is n factorial or n*(n-1)*(n-2) *… 1) (b) (16%) compute the probability of an even integer among the 100 integers: 1, 1+2, 1+2+3, 1+2+3+4, …., 1+2+3+… + 99, and 1+2+3+… + 100

In the following situation, determine whether you are asked to determine the number of permutations or combinations. then do the calculation how many ways are there to pick a starting five from a basketball team of twelve members? a. permutation; ps - 2520 b. combination; 1c, - 792 c. combination: 2cs - 2520 d. permutation; ps - 95040

Agarden consists of an apple tree, a pear tree, cauliflowers, and heads of cabbage. there are 40 vegetables in the garden. 24 of them are cauliflowers. what is the ratio of the number of cauliflowers to the number of heads of cabbage?