评估12 C 10
组合是一种计算结果总数的方法，其中结果的顺序无关紧要。这意味着可以按任何顺序选择对象。总而言之，任何两个整数的组合表示这些数字的所有可能的重新排列。例如，假设我们有三个名称，A、B 和 C，从这些名称中选择任意两个可能的名称，将对应于 AB、BC、AC。在组合中，AB 被认为等同于 BA。组合的总数可以通过以下公式获得，
C(n, r) = n!/(n – r)!r!
where,
n = total number of objects in the set
r = number of choosing objects from the set
A combination is an unordered permutation.
基本上有两种类型的组合：有重复的组合或不重复的组合 (r + n − 1)!r!(n − 1)! With repetition Without repetition Combinations n!r!(n − r)!
评估12 C 10
解决方案：
We know,
C(n, r) = n!/(n – r)!r!
Here, we have,
n = 12
r = 10
Putting the values of n and r in C(n, r) = n!/(n – r)!r!
We get,
C(12, 10) = 12!/(12 – 10)!10!
On solving, we get,
C(12, 10) = 12!/(2)!10!
Now,
C(12, 10) = 12 × 11 × 10!/2! × 10!
C(12, 10) = 12 × 11/ 2
On further simplifying, we get,
C(12, 10) = 132/2
C(12, 10) = 66
Therefore,
C(12,10). C(n,r) = n!/(n-r)!r! = 66
示例问题
问题 1. 求 C(6,4)。 C(n,r) = n!/(nr)!r!
解决方案：
Here,
n = 6
r = 4
Putting the values of n and r in C(n, r) = n!/(n – r)!r!
We get,
C(6, 4) = 6!/(6 – 4)!4!
C(6, 4) = 6!/(2)!4!
C(6, 4) = 6 × 5 × 4!/2! × 4!
C(6, 4) = 6 × 5/ 2
C(6, 4) = 30/2
C(6, 4) = 15
Therefore,
C(6, 4). C(n,r)=n!/(n-r)!r! = 15
问题 2. 评估 C(24, 20)。 C(n, r) = n!/(nr)!r!
解决方案：
Here,
n = 24
r = 20
Putting the values of n and r in C(n, r) = n!/(n – r)!r!
We get,
C(24, 20) = 24!/(24 – 20)!20!
C(24, 20) = 24!/(4)!20!
C(24, 20) = 24 × 23 × 22 × 21 × 20!/4! × 20!
C(24, 20) = 24 × 23 × 22 × 21/ 4 × 3 × 2 × 1
C(24, 20) = 255024/24
C(24, 20) = 10626
Therefore,
C(24, 20). C(n,r) = n!/(n-r)!r! = 10626
问题 3. 找出 C(18, 16)。 C(n, r) = n!/(nr)!r!
解决方案：
Here,
n = 18
r = 16
Putting the values of n and r in C(n, r) = n!/(n – r)!r!
We get,
C(18, 16) = 18!/(18 – 16)!16!
C(18, 16) = 18!/(2)!18!
C(18, 16) = 18 × 17 × 16!/2! × 16!
C(18, 16) = 24 × 23 × 22 × 21/ 2
C(24, 20) = 255024/24
C(24, 20) = 10626
Therefore,
C(24, 20). C(n,r) = n!/(n-r)!r! = 10626
问题 4. 排列与组合的区别？
解决方案：
Permutation CombinationArranging people, digits, numbers, alphabets, letters, and colours Selection of menu, food, clothes, subjects, team. Picking a team captain, pitcher and shortstop from a group. Picking three team members from a group. Picking two favourite colours, in order, from a colour brochure. Picking two colours from a colour brochure. Picking first, second and third place winners. Picking three winners.
问题 5. 从 9 个男孩和 12 个女孩中，找出 7 个男孩和 10 个女孩的一组中有多少个？
解决方案：
Here we have to find,
To select 7 boys out of 9 boys = 9C5 ways = 9!/(9-5)!5!
= 9!/4!5!
= 9 × 8 × 7 × 6 × 5!/4 × 3 × 2 × 1 × 5!
= 3024/24
= 126
To select 10 girls out of 12 girls = 12C10 ways = 12!/(12 – 10)!10!
= 12 × 11 × 10!/2!10!
= 132/2
= 66
Now,
The group can be selected in the 66 * 126 ways
= 8316 ways