Binomials and Lucas

Questions 1 to 5 of 8

1. 1. Lucas' theorem is run with modulus 6 and returns 0 for C(7, 2). What is the correct value of C(7, 2) modulo 6?

   0, because C(7, 2) is divisible by 63, because C(7, 2) = 21 and 21 leaves 3 modulo 61, because the base-6 digits of 7 and 2 are (1, 1) and (2, 0)21, because a modulus does not change the value
2. 2. What is C(24, 6) modulo 5?

   Type a number.

3. 3. Match each problem with the method the lesson recommends.

   Choose the matching item for each row.

   Modulus 6, and C(7, 2) is neededChoose…The factorial table from the previous lessonLucas' theorem, since the digit loop has at most about 60 stepsLucas' theorem with a table of 1000003 factorialsPascal's rule, because 6 is not primePrime 1000003, n = 10^18, many queriesChoose…The factorial table from the previous lessonLucas' theorem, since the digit loop has at most about 60 stepsLucas' theorem with a table of 1000003 factorialsPascal's rule, because 6 is not primePrime 1000000007, n = 10^6Choose…The factorial table from the previous lessonLucas' theorem, since the digit loop has at most about 60 stepsLucas' theorem with a table of 1000003 factorialsPascal's rule, because 6 is not primePrime 2, n = 10^18Choose…The factorial table from the previous lessonLucas' theorem, since the digit loop has at most about 60 stepsLucas' theorem with a table of 1000003 factorialsPascal's rule, because 6 is not prime
4. 4. An AI tool says that C(n, k) modulo 1000000 can be computed with factorials and modular inverses, because it is a common method. A user has n = 1000. What is the problem with this advice?

   Nothing is wrong, because factorial inverses always existThe modulus is not prime, so the inverses may not exist, and Pascal's rule is the safe choice for n = 1000Factorials only work when n is larger than the modulusThe table must use base-1000000 digits instead of factorials
5. 5. Lucas' theorem needs the modulus p to be what kind of number? Answer in one word.

   Type your answer.