A coin is tossed 10 times. Each outcome is a sequence of 10 heads and/or tails.

Question

A coin is tossed 10 times. Each outcome is a sequence of 10 heads and/or tails. What is the number of possible outcomes where the number of heads is exactly 4? At least 4? At most 4? Suppose that in a state, all car license plates consist of 6 characters where each character can be a capital letter or a decimal digit. How many different license plates are possible? How many possible plates where the first two characters are letters, and the last two characters are digits? How many plates that have just 3 letters are possible? A die is thrown 10 times, where the die has 6 faces labeled 1, 2, ..., 6. Each outcome will be a sequence of 10 faces (i.e., 10 digits), where each face is one of the 6 values. What is the number of possible outcomes where face 2 comes up exactly 3 times? What is the number of possible outcomes where face 2 comes up exactly 2 times and face 3 comes up exactly 3 times? One urn contains 20 balls: 5 balls are red (labeled R_1, R_2, .... R_5), 6 are blue (labeled B_1, B_2, .... B_6), and 9 are white (labeled W_1, W_2, ..., W_9). You draw 8 balls from the urn. What is the number of possible outcomes where 2 of the drawn balls are red, 3 are blue, and 3 are white?

Explanation / Answer

e)

exactly 4 head = 10 C4 =210

at lest 4 head = 10C4+10C5+10C6+10C7+10C8+10C9+10C10=848

at most 4 head = 10C0 + 10C1+10C2+10C3+10C4=386

f ) 26 capital latter and 10 digit out of that 6 character is in licence plate

i) out of (26+10=36) 6 character required to form number plate

number of way possible is 36 C 6= 1947792

2) first 2 are letter and last 2 are number = 26 C 2 * 10 C 2 * 32 C 2 = 7254000

3) only 3 letters = 26C3 * 10 C3 = 312000

h )

( 2 red and 3 blue and 3 white ) =5 C 2 *6 C 3 * 9 C 3 = 10* 20*84=16800

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