the binomial 1.2-8 Pascal\'s triangle gives a method for cakculating follows: 1
ID: 3041117 • Letter: T
Question
the binomial 1.2-8 Pascal's triangle gives a method for cakculating follows: 1 5 10 10 S The ath row of this triangle gives the coefficients for (a +by-1, To find an the table other than a 1 on the boundary, add the two nearest numbers in the n directly above. The equation called Pascal's equation, explains why Pascal's triangle works. Prove that ths equation is correct. 12.9 Among nine presidential candidates at a debate, three are Republicans and six are tS. (a) In how many different ways can the nine candidates be lined up? b) How many lineups by party are possible if each candidate is labeled either (e) For each of the nine candidates, you 'are to decide whether the candidate did a good job or a poor job; that is, give each of the nine candidates a grade of Gr P. How many different "score cards" are possible? 1.2-10 Prove -10 and HINT: Consider (1-1)t and (1 + 1)t or use Pascal's equation and proof by induc- tion. 12-11 A poker hand is defined as drawing five cards at random without replacement from a deck of 52 playing cards. Find the probability of each of the following poker hands (four cards of equal face value and one card of a different value). (b) Full house (one pair and one triple of cards with equal face value). e) Three of a kind (three equal face values plus two cards of different values). (d) Two pairs (two pairs of equal face value plus one card of a different value). (e) One pair (one pair of equal face value plus three cards of different values)Explanation / Answer
Solution 1-2-9:
a. Number of ways = 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880
b. Number of lineups by party = 9C3= 9!/3!*6! = 84
c. Since there are two choices for each candidate G or P, so the number of different scorecards are 2^9 = 512
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