7. In how many ways can 10 men be placed side by side? 8. How many different words can be formed out of the letters in Binomial?

9. How many different permutations can be formed out of the letters in Algebra taken all together?

10. The number of things: number of permutations of the things taken 3 at a time: 1:20. How many things are there?

=

11. The number of permutations of m things taken 3 at a time: the number of permutations of m+2 things taken 3 at a time=1:5. Find m.

12. In the permutations of a, b, c, d, e, f, g taken all together, find how many begin with cd.

13. Find the number of permutations of the letters of the product a2b3c4 written at full length.

14. Find the number of permutations that can be formed out of the letters in each of the following words: Conceit, Talavera, Calcutta, Proposition, Mississippi.

XXXIV. COMBINATIONS.

407. THE COMBINATIONS of a number of things are the different collections that can be formed out of them by taking a certain number at a time, without regard to the order in which the things stand in each collection.

Thus the combinations of a, b, c, d taken two at a time are ab, ac, ad, bc, bd, cd.

Here from each combination we could make two permutations: thus ab, ba; ac, ca; and so on: for ab, ba are the same combination, and so are ac, ca.

Similarly the combinations of a, b, c, d taken three at a time are abc, abd, acd, bcd.

Here from each combination we could make six permutations; thus abc, acb, bac, bça, cab, cba: and so on.

P

from

And, generally, in accordance with Art. 405, any combination of n things may be made into 1.2.3... n permutations.

408. To find the number of combinations of n different things taken r at a time.

Let C, denote the number of combinations required.

Since each combination contains r things it can be made intor permutations (Art. 405);

.. the whole number of permutations=[r. C..

But also from (Art. 404) the whole number of permutations of n things taken r at a time

409. To shew that the number of combinations of n things taken r at a time is the same as the number taken n-r at a time.

n.

In

1.2.3

...... r

X

1.2.3
n. (n-1).

......

(n-r)
. (r+1)

.(n-1) (n− r + 1). (n − r). 3.2.1

1.2.3 r. (r+1).

......

(n-1). n

410. Making r= 1, 2, 3 ............... r −1, r, r+1 in order,

......

Hence the general expression for the factor connecting C,, one of the set of numbers C1, C2...... Cr+1 .On, with

C-1, that which stands next before it, is

n-r+1
r

n-r+1
r

that is,

n-r+1

With regard to this factor

we observe

r

(1) It is always positive, because n+1 is greater than r.

(2) Its value continually decreases, for

successive values of r it is greater than unity, and therefore each of the corresponding terms is greater than the preceding.

(4) When r is such that

n-r+1
r

is less than unity the

corresponding term is less than the preceding.

(5) If n and r be such that

n−1 + r
r

=

=1, C, and C-1 are

a pair of equal terms, each greater than any preceding or subsequent term.

Hence up to a certain term (or pair of terms) the terms increase, and after that decrease: this term (or pair of terms) is the greatest of the series, and it is the object of the next Article to determine what value of r gives this greatest term (or pair of terms).

411. To find the value of r for which the number of combinations of n things taken r together is the greatest.

Hence, if C, denote the number of combinations required,

or, n-r+1 is not less than r and r+1 not less than n−r,

or,

n+1 is not less than 2r and 2r not less than n−1; .. 2r is not greater than n+1 and not less than n−1. Hence 2r can have only three values, n−1, n, n + 1. Now 2r must be an even number, and therefore

(1) If n be odd, n-1 and n+1 being both even numbers, 2r may be equal to n-1 or n + 1;

(2) If n be even, n−1 and n+1 being both odd numbers, 2r can only be equal to n;

Ex. (1) Of eight things how many must be taken together that the number of combinations may be the greatest possible?

Here n=8, an even number, therefore the number to be 8 × 7×6×5

taken is 4, which will give

1 × 2 × 3 × 4

or 70 combinations.

Ex. (2) If the number of things be 9, then the number to

1. Out of 100 soldiers how many different parties of 4 can be chosen ?

2. How many combinations can be made of 6 things taken 5 at a time?

3. Of the combinations of the first 10 letters of the alphabet taken 5 together, in how many will a occur?

4. How many words can be formed, consisting of 3 consonants and one vowel, in a language containing 19 consonants and 5 vowels?

5. The number of combinations of n things taken 4 at a time the number taken 2 at a time 15: 2. Find n.

6.

The number of combinations of n things, taken 5 at