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The

number of ways that 4 marbles can be chosen from 21 different
marbles is equal to the number of ways that m marbles can be chosen
from the same 21 marbles. if m≠4 then the value of m is?

User Nayara
by
8.2k points

1 Answer

1 vote

Answer:

m = 18

Step-by-step explanation:

To solve this problem, we need to find the value of m such that the number of ways to choose 4 marbles is equal to the number of ways to choose m marbles from a set of 21 marbles.

The number of ways to choose k items from a set of n items is given by the binomial coefficient, also known as "n choose k," which is denoted as C(n, k).

In this case, the number of ways to choose 4 marbles from 21 marbles is C(21, 4), and the number of ways to choose m marbles from the same 21 marbles is C(21, m).

We are given that C(21, 4) = C(21, m).

Using the formula for binomial coefficients, we have:

C(21, 4) = C(21, m)

21! / (4! * (21-4)!) = 21! / (m! * (21-m)!)

Simplifying further:

(21! * m! * (21-m)!) / (4! * (21-4)!) = 1

Cancelling out the common terms:

(m! * (21-m)!) / (4! * (21-4)!) = 1

Simplifying the factorials:

(m! * (21-m)!) / (4! * 17!) = 1

(m! * (21-m)!) = (4! * 17!)

Since factorials are always positive, we can remove the factorials from both sides:

(m * (m-1) * ... * 1) * ((21-m) * (21-m-1) * ... * 1) = (4 * 3 * 2 * 1) * (17 * 16 * ... * 1)

Cancelling out the common terms:

(m * (m-1) * ... * 1) * ((21-m) * (21-m-1) * ... * 1) = (4 * 3 * 2 * 1) * (17 * 16 * ... * 1)

Expanding the products:

m! * (21-m)! = 24 * 17!

We know that 24 = 4 * 6, so we can rewrite the equation as:

m! * (21-m)! = (4 * 6) * 17!

We see that 6 is a factor in both m! and (21-m)!, so we can simplify further:

(6 * (m! / 6) * ((21-m)! / 6)) = 4 * 17!

Simplifying:

(m-1)! * ((21-m)! / 6) = 4 * 17!

Since 17! does not have a factor of 6, we know that (21-m)! / 6 must equal 1:

(21-m)! / 6 = 1

Solving for (21-m)!, we have:

(21-m)! = 6

The only positive integer value of (21-m)! that equals 6 is (21-m)! = 3.

Therefore, (21-m) = 3, and solving for m:

21 - m = 3

m = 21 - 3

m = 18

Thus, the value of m is 18.

User Dubilla
by
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