Probability - Gravner - 14

Question

A California license plate consists of a sequence of seven symbols: number, letter, letter, letter, number, number, number, where a letter is any one of $26$ letters and a number is one among $0, 1, . . . , 9$. Assume that all license plates are equally likely.

(a) What is the probability that all symbols are different?

(b) What is the probability that all symbols are different and the first number is the largest among the numbers?

Comments

a) $0.447337$

b) $\large \frac{26*25*24( 9*8*7 + 8*7*6 + 7*6*5 + 6*5*4+ 5*4*3 + 4*3*2 + 3*2*1 )}{10^4*26^3}$$ = 0.11183432$