Question

Let A be the set of all strings of decimal digits of length five. For example 00312 and 19483 are strings in A.

a. How many strings in A begin with 774?

b. How many strings in A have exactly one 8?

c. How many strings in A have exactly three 6’s?

d. How many strings in A have the digits in a strictly increasing order? For example 02357 and 14567 are such strings, but 31482 and 12335 are not.

Answer 1

Answer ;

a. Since the string begins with 774, we need to bother about the last two digits.

Each digit has 10 choices and so there are 10 * 10 = 100 such strings.

b. The one 8 can be in any of the 5 places. The remaining 4 places have 94 choices.

Therefore, total number of such strings = 5 * 94 = 32805.

c. The three 6's can occupy their places in 5C3 = (5 4) / (2 1) = 10 ways.

The remaining two digits have 92 = 81 choices.

Total number of such strings = 10 * 81 = 810.

d. The question is actually chosing 5 digits out of 10 without worrying about order (the order is fixed i.e. ascending).

This can be done in 10C5 = 10! / (5! * 5!) = 252 ways.

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